Difference between revisions of "Non-self-adjoint operator"
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− | + | A linear operator in a Hilbert space the spectral analysis of which cannot be made to fit into the framework of the theory of self-adjoint operators (cf. [[Self-adjoint operator|Self-adjoint operator]]) and its simplest generalizations: the theory of unitary operators (cf. [[Unitary operator|Unitary operator]]) and the theory of normal operators (cf. [[Normal operator|Normal operator]]). Non-self-adjoint operators arise in the discussion of processes that proceed without conservation of energy: in problems with friction, in the theory of open resonators, in problems of inelastic scattering, and others. Certain self-adjoint problems, in which by separation of variables an operator-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n0672801.png" /> appears that depends non-linearly on a spectral parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n0672802.png" />, also lead to a study of non-self-adjoint operators. Many of the propositions referring to the theory of non-self-adjoint operators are valid also for operators acting in arbitrary Banach spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n0672803.png" />-spaces, topological vector spaces, etc. | |
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− | A linear operator in a Hilbert space the spectral analysis of which cannot be made to fit into the framework of the theory of self-adjoint operators (cf. [[Self-adjoint operator|Self-adjoint operator]]) and its simplest generalizations: the theory of unitary operators (cf. [[Unitary operator|Unitary operator]]) and the theory of normal operators (cf. [[Normal operator|Normal operator]]). Non-self-adjoint operators arise in the discussion of processes that proceed without conservation of energy: in problems with friction, in the theory of open resonators, in problems of inelastic scattering, and others. Certain self-adjoint problems, in which by separation of variables an operator-valued function | ||
− | appears that depends non-linearly on a spectral parameter | ||
− | also lead to a study of non-self-adjoint operators. Many of the propositions referring to the theory of non-self-adjoint operators are valid also for operators acting in arbitrary Banach spaces, | ||
− | spaces, topological vector spaces, etc. | ||
The most extensive method in the study of non-self-adjoint operators is that of evaluating the [[Resolvent|resolvent]], which makes use of the theory of analytic functions, of asymptotic expansions, etc. | The most extensive method in the study of non-self-adjoint operators is that of evaluating the [[Resolvent|resolvent]], which makes use of the theory of analytic functions, of asymptotic expansions, etc. | ||
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In the development of a general theory of non-self-adjoint operators (in particular, partial differential operators) an important role was played by the work of M.V. Keldysh [[#References|[1]]] (see also [[#References|[2]]]). He studied an equation of the form | In the development of a general theory of non-self-adjoint operators (in particular, partial differential operators) an important role was played by the work of M.V. Keldysh [[#References|[1]]] (see also [[#References|[2]]]). He studied an equation of the form | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n0672804.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table> | |
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− | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n0672805.png" /> is an element of a certain Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n0672806.png" /> and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n0672807.png" /> has the representation | |
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− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n0672808.png" /></td> </tr></table> | |
− | + | Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n0672809.png" /> is a completely-continuous invertible self-adjoint operator of finite order and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728011.png" />, are arbitrary completely-continuous operators. (A completely-continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728012.png" /> acting on a Hilbert space is said to be an operator of finite order if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728013.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728015.png" />; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728016.png" /> denote the singular numbers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728017.png" />, that is, the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728018.png" />.) The eigen values of (1) are those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728019.png" /> for which the equation has non-trivial solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728020.png" />; these solutions are called eigen vectors. | |
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− | + | Under the assumptions made above the spectrum of (1) is discrete. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728021.png" /> is non-self-adjoint, apart from the eigen vectors there naturally arise (in the presence of a multiple spectrum) associated vectors. In [[#References|[1]]] a chain is constructed of associated vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728022.png" /> corresponding to an eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728023.png" /> and eigen vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728024.png" /> according to the rule | |
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table> | |
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− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728026.png" /></td> </tr></table> | |
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− | The system of eigen vectors and associated vectors of | + | The system of eigen vectors and associated vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728027.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728029.png" />-fold complete if any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728030.png" /> vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728032.png" /> can be approximated in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728033.png" /> with arbitrary accuracy by finite linear combinations of the form |
− | is said to be | ||
− | fold complete if any | ||
− | vectors | ||
− | of | ||
− | can be approximated in the norm of | ||
− | with arbitrary accuracy by finite linear combinations of the form | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728034.png" /></td> </tr></table> | |
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− | + | with the same coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728035.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728036.png" /> a is vector-valued function of the form | |
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728037.png" /></td> </tr></table> | |
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− | with the | + | The definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728038.png" />-fold completeness is, naturally, connected with the solution of the Cauchy problem for the non-stationary equation corresponding to (1). |
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− | + | According to a theorem of Keldysh, under the assumptions made on the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728039.png" /> the system of all eigen vectors and associated vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728040.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728041.png" />-fold complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728042.png" />. In [[#References|[1]]] he also proved that the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728043.png" /> can be approximated asymptotically on rays with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728044.png" />. In the proof of completeness Keldysh developed a new method for evaluating the resolvent of an abstract completely-continuous non-self-adjoint operator of finite order. Here emerged the special role played in the completeness problem by Volterra operators, i.e. completely-continuous operators with a single point of the spectrum at zero. In establishing the asymptotic behaviour of the eigen values, Keldysh [[#References|[3]]] used a new Tauberian theorem due to him. | |
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− | + | Keldysh's research was continued by many authors. His theorem was extended in [[#References|[4]]] to the case when the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728045.png" /> depends rationally on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728046.png" />. | |
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− | + | In [[#References|[5]]]–[[#References|[7]]] the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728047.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728048.png" /> is considered, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728049.png" /> is a completely-continuous positive-definite operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728050.png" /> is a bounded self-adjoint operator. A generalization of Pontryagin's theorem (see [[#References|[8]]]) on the existence of a maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728051.png" />-non-negative invariant subspace for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728052.png" />-self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728053.png" /> made it possible (see [[#References|[6]]] and [[#References|[7]]]) to establish the two-fold completeness of all eigen vectors and associated vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728054.png" /> in situations important for applications, and also the one-fold completeness of the subsystem corresponding to the spectrum situated in the left (or right) half-plane. These results have been developed much further. | |
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− | In [[#References|[ | ||
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− | + | The summability of Fourier series in eigen vectors and associated vectors of a completely-continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728055.png" /> of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728056.png" /> has been established (see [[#References|[9]]]) if the eigen values of the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728057.png" /> lie in a sector of the complex plane with opening angle less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728058.png" />. (For applications of this theorem and further generalizations of it see [[#References|[10]]] and the references there.) | |
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− | The summability of Fourier series in eigen vectors and associated vectors of a completely-continuous operator | ||
− | of finite order | ||
− | has been established (see [[#References|[9]]]) if the eigen values of the quadratic form | ||
− | lie in a sector of the complex plane with opening angle less than | ||
− | (For applications of this theorem and further generalizations of it see [[#References|[10]]] and the references there.) | ||
The question when the system of eigen vectors and associated vectors forms a basis in the Hilbert space has been studied in a number of papers. Most general conditions under which the system of eigen vectors and associated vectors of a dissipative completely-continuous operator forms a basis are found in [[#References|[12]]]. | The question when the system of eigen vectors and associated vectors forms a basis in the Hilbert space has been studied in a number of papers. Most general conditions under which the system of eigen vectors and associated vectors of a dissipative completely-continuous operator forms a basis are found in [[#References|[12]]]. | ||
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In the case of singular differential operators with discrete spectrum a number of subtle results have been obtained (see [[#References|[11]]] and [[#References|[13]]]) on the completeness of the eigen functions and associated functions of a Sturm–Liouville operator with complex potential. Important results have been obtained for the case of an elliptic operator (see [[#References|[14]]]). Keldysh's theorem has been generalized to the case of generalized eigen functions and associated functions of non-self-adjoint elliptic operators (see [[#References|[15]]], [[#References|[16]]]). | In the case of singular differential operators with discrete spectrum a number of subtle results have been obtained (see [[#References|[11]]] and [[#References|[13]]]) on the completeness of the eigen functions and associated functions of a Sturm–Liouville operator with complex potential. Important results have been obtained for the case of an elliptic operator (see [[#References|[14]]]). Keldysh's theorem has been generalized to the case of generalized eigen functions and associated functions of non-self-adjoint elliptic operators (see [[#References|[15]]], [[#References|[16]]]). | ||
− | An attempt to carry over the theorem on the reduction to Jordan form of a finite-dimensional operator to the infinite-dimensional case has led to the construction of a triangular integral representation. For completely-continuous operators | + | An attempt to carry over the theorem on the reduction to Jordan form of a finite-dimensional operator to the infinite-dimensional case has led to the construction of a triangular integral representation. For completely-continuous operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728059.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728061.png" /> are self-adjoint and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728062.png" /> is of finite order, an analogue has been obtained of Schur's theorem on the unitary equivalence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728063.png" /> to a triangular operator (see [[#References|[17]]]). A special place in the problem of triangular representation is occupied by Volterra operators. |
− | where | ||
− | and | ||
− | are self-adjoint and | ||
− | is of finite order, an analogue has been obtained of Schur's theorem on the unitary equivalence of | ||
− | to a triangular operator (see [[#References|[17]]]). A special place in the problem of triangular representation is occupied by Volterra operators. | ||
− | A special role in this problem is played by von Neumann's theorem, stating that a completely-continuous linear operator in a Hilbert space has a non-trivial invariant subspace; an arbitrary bounded linear operator in a Banach space need not have an invariant subspace; the corresponding problem for the case of a Hilbert space is still open (1989). A Volterra operator is said to be a unicellular operator if for any two invariant subspaces | + | A special role in this problem is played by von Neumann's theorem, stating that a completely-continuous linear operator in a Hilbert space has a non-trivial invariant subspace; an arbitrary bounded linear operator in a Banach space need not have an invariant subspace; the corresponding problem for the case of a Hilbert space is still open (1989). A Volterra operator is said to be a unicellular operator if for any two invariant subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728065.png" /> of it either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728066.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728067.png" />. In [[#References|[18]]] a necessary and sufficient condition was found for an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728068.png" /> to be a unicellular operator under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728069.png" /> is nuclear and non-negative definite; this condition can be stated in terms of the growth of the resolvent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728070.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728071.png" />. A simple sufficient condition for being a unicellular operator is indicated in [[#References|[19]]]. |
− | and | ||
− | of it either | ||
− | or | ||
− | In [[#References|[18]]] a necessary and sufficient condition was found for an operator | ||
− | to be a unicellular operator under the assumption that | ||
− | is nuclear and non-negative definite; this condition can be stated in terms of the growth of the resolvent of | ||
− | as | ||
− | A simple sufficient condition for being a unicellular operator is indicated in [[#References|[19]]]. | ||
Non-self-adjoint operators with a continuous spectrum were first studied by M.A. Naimark (see [[#References|[20]]], [[#References|[21]]]), who obtained an expansion in a Fourier integral, connected with the non-self-adjoint problem | Non-self-adjoint operators with a continuous spectrum were first studied by M.A. Naimark (see [[#References|[20]]], [[#References|[21]]]), who obtained an expansion in a Fourier integral, connected with the non-self-adjoint problem | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728072.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table> | |
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− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table> | |
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− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728074.png" /> is a complex-valued function subject to the condition |
− | is a complex-valued function subject to the condition | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728075.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table> | |
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− | ( | ||
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− | and | + | and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728076.png" /> is a complex number. The results in [[#References|[20]]] imply, in particular, that in a neighbourhood of points of the real axis where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728077.png" /> vanishes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728078.png" />, the spectral projections of the operator (3)–(4) are unbounded. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728079.png" /> denotes the solution of (3) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728080.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728081.png" />, the Jost solution.) In [[#References|[20]]], the real zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728082.png" /> are called spectral singularities. In [[#References|[22]]] a generalization of the results of [[#References|[20]]] to the case of the Schrödinger equation in the three-dimensional space is obtained. In a further development of [[#References|[20]]] it has been proved (see [[#References|[23]]]) that in general (without a restriction of type (5)) the spectral function of a differential operator must be regarded as a continuous linear functional on a certain topological space. |
− | is a complex number. The results in [[#References|[20]]] imply, in particular, that in a neighbourhood of points of the real axis where | ||
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− | the spectral projections of the operator (3)–(4) are unbounded. (Here | ||
− | denotes the solution of (3) for which | ||
− | as | ||
− | the Jost solution.) In [[#References|[20]]], the real zeros of | ||
− | are called spectral singularities. In [[#References|[22]]] a generalization of the results of [[#References|[20]]] to the case of the Schrödinger equation in the three-dimensional space is obtained. In a further development of [[#References|[20]]] it has been proved (see [[#References|[23]]]) that in general (without a restriction of type (5)) the spectral function of a differential operator must be regarded as a continuous linear functional on a certain topological space. | ||
In [[#References|[24]]] a system of non-self-adjoint equations is studied with singular points whose position depends on the spectral parameter. These systems occur in the theory of shells without moments (cf. [[Shell theory|Shell theory]]). For such systems asymptotic properties of the solutions have been established, and also solvability theorems of Cauchy type have been proved. Completeness theorems for the system of eigen functions and associated functions of non-self-adjoint integro-differential operators generating a non-regular problem have been established. | In [[#References|[24]]] a system of non-self-adjoint equations is studied with singular points whose position depends on the spectral parameter. These systems occur in the theory of shells without moments (cf. [[Shell theory|Shell theory]]). For such systems asymptotic properties of the solutions have been established, and also solvability theorems of Cauchy type have been proved. Completeness theorems for the system of eigen functions and associated functions of non-self-adjoint integro-differential operators generating a non-regular problem have been established. | ||
Line 193: | Line 65: | ||
An important problem in the theory of non-self-adjoint operators is the expansion of the kernel of the Green operator of a biorthogonal series in eigen functions and associated functions and also the question of a basis. Tamarkin [[#References|[25]]] has studied the expansion of a summable function in a series of eigen functions and associated functions of a regular problem, and also the question of equiconvergence with a trigonometric Fourier series. Later it was proved (see , [[#References|[27]]]) that for non-regular problems equiconvergence with a trigonometric series does not hold. | An important problem in the theory of non-self-adjoint operators is the expansion of the kernel of the Green operator of a biorthogonal series in eigen functions and associated functions and also the question of a basis. Tamarkin [[#References|[25]]] has studied the expansion of a summable function in a series of eigen functions and associated functions of a regular problem, and also the question of equiconvergence with a trigonometric Fourier series. Later it was proved (see , [[#References|[27]]]) that for non-regular problems equiconvergence with a trigonometric series does not hold. | ||
− | For strongly-regular conditions the system of eigen functions and associated functions forms a basis in | + | For strongly-regular conditions the system of eigen functions and associated functions forms a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728083.png" />. It has been proved (see [[#References|[28]]], [[#References|[29]]]) that this system not only forms a basis, but even a so-called Riesz basis. |
− | It has been proved (see [[#References|[28]]], [[#References|[29]]]) that this system not only forms a basis, but even a so-called Riesz basis. | ||
In [[#References|[30]]] an important method has been developed for studying the basis property and the uniform convergence of the expansions in eigen functions and associated functions of an ordinary non-self-adjoint operator. This method is a further development of the ideas applied in the investigation of self-adjoint problems. A new treatment for the eigen functions and associated functions has been proposed, making it possible to dispense with the specific form of the boundary conditions; general ordinary differential operators or pencils of such operators are considered and necessary and sufficient conditions have been established for the basis property of the eigen functions and associated functions of such operators, as well as a criterion for equiconvergence. The method is based only on a mean value formula for the eigen functions and associated functions (see also [[#References|[31]]]). It also turns out that if the operator has infinitely many associated functions, then the basis property depends on the choice of the latter (see [[#References|[32]]]). | In [[#References|[30]]] an important method has been developed for studying the basis property and the uniform convergence of the expansions in eigen functions and associated functions of an ordinary non-self-adjoint operator. This method is a further development of the ideas applied in the investigation of self-adjoint problems. A new treatment for the eigen functions and associated functions has been proposed, making it possible to dispense with the specific form of the boundary conditions; general ordinary differential operators or pencils of such operators are considered and necessary and sufficient conditions have been established for the basis property of the eigen functions and associated functions of such operators, as well as a criterion for equiconvergence. The method is based only on a mean value formula for the eigen functions and associated functions (see also [[#References|[31]]]). It also turns out that if the operator has infinitely many associated functions, then the basis property depends on the choice of the latter (see [[#References|[32]]]). | ||
Line 200: | Line 71: | ||
For non-self-adjoint elliptic operators there is convergence (see [[#References|[33]]]) of a certain sequence of means of Poisson type of the partial sums of the biorthogonal series, that is, a summation method has been proposed. | For non-self-adjoint elliptic operators there is convergence (see [[#References|[33]]]) of a certain sequence of means of Poisson type of the partial sums of the biorthogonal series, that is, a summation method has been proposed. | ||
− | Expansions in eigen functions and associated functions of non-regular problems were first obtained for problems of the form | + | Expansions in eigen functions and associated functions of non-regular problems were first obtained for problems of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728085.png" />. It has been proved (see ) that expansions in uniformly convergent series of this form can hold for functions satisfying a certain analyticity condition. |
− | |||
− | It has been proved (see ) that expansions in uniformly convergent series of this form can hold for functions satisfying a certain analyticity condition. | ||
Among the fundamental studies is the problem (see [[#References|[34]]]) in which the perturbation of the spectrum of the Laplace operator under a change of domain is studied. Here for the first time the role of the capacity of the set which is varying on the spectrum of the operator became manifest. These methods were successfully applied in the study of non-self-adjoint operators. | Among the fundamental studies is the problem (see [[#References|[34]]]) in which the perturbation of the spectrum of the Laplace operator under a change of domain is studied. Here for the first time the role of the capacity of the set which is varying on the spectrum of the operator became manifest. These methods were successfully applied in the study of non-self-adjoint operators. | ||
Line 214: | Line 83: | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.V. Keldysh, "On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations" ''Dokl. Akad. Nauk SSSR'' , '''77''' : 1 (1951) pp. 11–14 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.V. Keldysh, "On the completeness of the eigenfunctions of some class of non-selfadjoint linear operators" ''Russian Math. Surveys'' , '''26''' : 4 (1971) pp. 15–44 ''Uspekhi Mat. Nauk'' , '''26''' : 4 (1971) pp. 15–41</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.V. Keldysh, "On a Tauberian theorem" ''Trudy Mat. Inst. Steklov.'' , '''38''' (1951) pp. 77–86 (In Russian) {{MR|}} {{ZBL|0272.40007}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D.E. Allakhverdiev, "On completeness of a system of eigenelements and associated elements of a nonselfadjoint operator close to normal" ''Dokl. Akad. Nauk SSSR'' , '''115''' : 2 (1957) pp. 207–210 (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H.K. Langer, "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728086.png" />-Hermitian operators" ''Soviet Math. Dokl.'' , '''1''' : 4 (1960) pp. 1052–1055 ''Dokl. Akad. Nauk SSSR'' , '''134''' : 2 (1960) pp. 263–266</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.G. Krein, "A new application of the fixed-point principle in the theory of operators on a space with indefinite metric" ''Soviet Math. Dokl.'' , '''5''' : 1 (1964) pp. 266–269 ''Dokl. Akad. Nauk SSSR'' , '''154''' : 5 (1964) pp. 1023–1026 {{MR|169059}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> H.K. Langer, M.G. Krein, "Applications of function theory in the mechanics of a continuous medium" , Moscow (1965) (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> L.S. Pontryagin, "Hermitian operators in spaces with indefinite metric" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''8''' (1944) pp. 243–280 (In Russian) {{MR|}} {{ZBL|0061.26004}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> V.B. Lidskii, "On completeness of a system of eigenelements and associated functions of a nonselfadjoint operator" ''Dokl. Akad. Nauk. SSSR'' , '''110''' : 2 (1956) pp. 172–175 (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M.S. Agranovich, "Summability of series in root vectors of non-self-adjoint elliptic operators" ''Funct. Anal. Appl.'' , '''10''' : 3 (1976) pp. 1–3 ''Funktsional. Anal. i Prilozhen.'' , '''10''' : 3 (1976) pp. 1–12 {{MR|}} {{ZBL|0362.35062}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , ''Transl. Math. Monogr.'' , '''18''' , Amer. Math. Soc. (1969) (Translated from Russian) {{MR|0246142}} {{ZBL|0181.13504}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> V.E. Katsnel'son, "Conditions under which systems of eigenvectors of some classes of operators form a basis" ''Funct. Anal. Appl.'' , '''1''' : 2 (1967) pp. 122–133 ''Funksional. Anal. i Prilozhen.'' , '''1''' : 2 (1967) pp. 39–51</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> B.V. Lidskii, "Conditions for complete continuity of the resolvent of a differential operator" ''Dokl. Akad. Nauk SSSR'' , '''113''' : 1 (1957) pp. 28–31 (In Russian) {{MR|0091385}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A.G. Kostyuchenko, "Asymptotic distribution of the eigenvalues of elliptic operators" ''Soviet Math. Dokl.'' , '''5''' : 5 (1964) pp. 1171–1175 ''Dokl. Akad. Nauk. SSSR'' , '''158''' : 1 (1964) pp. 41–44</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> S.M. Ponomarev, "A generalization of M.V. Keldysh's theorem concerning the completeness of eigen- and associated functions of the first boundary value problem for a nonself-adjoint elliptic operator" ''Diff. Eq.'' , '''10''' : 12 (1974) pp. 1777–1779 ''Differentsial. Uravnen.'' , '''10''' : 12 (1974) pp. 2294–2296</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> N.M. Krukovskii, "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728087.png" />-tuple completeness of the system of generalized eigenfunctions and associated functions of a non-self-adjoint elliptic operator" ''Diff. Eq.'' , '''12''' : 10 (1976) pp. 1290–1296 ''Differentsial. Uravnen.'' , '''12''' : 10 (1976) pp. 1832–1851</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Theory and applications of Volterra operators in Hilbert space" , Amer. Math. Soc. (1970) (Translated from Russian) {{MR|0264447}} {{ZBL|0194.43804}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> M.S. Brodskii, G.E. Kisilevskii, "Criterion for unicellularity of dissipative Volterra operators with nuclear imaginary components" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''30''' : 6 (1966) pp. 1213–1228 (In Russian) {{MR|0203459}} {{ZBL|}} </TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> N.K. Nikol'skii, "The unicellularity and nonunicellularity of weighted shift operators" ''Soviet Math. Dokl.'' , '''8''' : 1 (1967) pp. 91–94 ''Dokl. Akad. Nauk SSSR'' , '''172''' : 2 (1967) pp. 287–290 {{MR|}} {{ZBL|0157.21202}} </TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> M.A. Naimark, "Investigation of the spectrum and the expansion in eigenfunctions of a second-order non-self-adjoint operator on a semi-axis" ''Trudy Moskov. Mat. Obshch.'' , '''3''' (1954) pp. 181–270 (In Russian)</TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top"> M.A. Naimark, "Linear differential operators" , '''1–2''' , Harrap (1968) (Translated from Russian) {{MR|0262880}} {{ZBL|0227.34020}} </TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top"> M.G. Gasymov, "Expansion in solutions of the scattering problem for a nonselfadjoint Schrödinger equation" ''Dokl. Akad. Nauk AzSSR'' , '''22''' : 10 (1966) pp. 9–12 (In Russian)</TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top"> V.A. Marchenko, "Eigenfunction expansions of non-self-adjoint singular second-order differential operators" ''Mat. Sb.'' , '''52''' (1960) pp. 739–788 (In Russian)</TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top"> V.A. Sadovnichii, "Analytic properties of solutions of system of equations with singularities that form segments" ''Trudy Sem. Petrov.'' , '''2''' (1976) pp. 211–221 (In Russian)</TD></TR><TR><TD valign="top">[25]</TD> <TD valign="top"> Ya.D. Tamarkin, "On some general problems of the theory of ordinary linear differential equations..." , Petrograd (1917) (In Russian)</TD></TR><TR><TD valign="top">[26a]</TD> <TD valign="top"> L.E. Ward, "An irregular boundary value and expansion problem" ''Ann. of Math. (2)'' , '''26''' (1925) pp. 21–36 {{MR|1502673}} {{ZBL|50.0314.03}} </TD></TR><TR><TD valign="top">[26b]</TD> <TD valign="top"> L.E. Ward, "A third order irregular boundary value problem and the associated series" ''Trans. Amer. Math. Soc.'' , '''34''' (1932) pp. 417–434 {{MR|1501646}} {{ZBL|0004.11001}} {{ZBL|58.0467.02}} </TD></TR><TR><TD valign="top">[27]</TD> <TD valign="top"> A.P. Khromov, , ''Proc. 2-nd Conf. Inst. Higher Education'' , '''1''' , Kuibyshev (1962) pp. 109–113 (In Russian)</TD></TR><TR><TD valign="top">[28]</TD> <TD valign="top"> V.P. Mikhailov, "Riesz bases in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728088.png" />" ''Soviet Math. Dokl.'' , '''3''' : 3 (1962) pp. 851–854 ''Dokl. Akad. Nauk SSSR'' , '''144''' : 5 (1962) pp. 981–984</TD></TR><TR><TD valign="top">[29]</TD> <TD valign="top"> G.M. Kessel'man, ''Izv. Vyzov. Mat.'' , '''2''' (1964) pp. 82–93</TD></TR><TR><TD valign="top">[30]</TD> <TD valign="top"> V.A. Il'in, "On the properties of a reduced system of eigenfunctions and associated functions of a Keldysh bundle of ordinary differential operators" ''Soviet Math. Dokl.'' , '''17''' : 5 (1976) pp. 1247–1250 ''Dokl. Akad. Nauk SSSR'' , '''230''' (1976) pp. 30–33</TD></TR><TR><TD valign="top">[31]</TD> <TD valign="top"> E.N. Moisev, "The uniform convergence of certain expansions in a closed domain" ''Soviet Math. Dokl.'' , '''18''' : 2 (1977) pp. 549–553 ''Dokl. Akad. Nauk SSSR'' , '''233''' (1977) pp. 1042–1045</TD></TR><TR><TD valign="top">[32]</TD> <TD valign="top"> V.V. Tikhomirov, "On the Riesz means of expansions in eigenfunctions and associated functions of a nonselfadjoint ordinary differential operator" ''Math. USSR Sb.'' , '''31''' : 1 (1977) pp. 29–48 ''Mat. Sb.'' , '''102''' : 1 (1977) pp. 33–55 {{MR|}} {{ZBL|0388.47027}} </TD></TR><TR><TD valign="top">[33]</TD> <TD valign="top"> V.B. Lidskii, "Fourier series expansion of principal functions of a non-self-adjoint elliptic operator" ''Mat. Sb.'' , '''57''' : 2 (1962) pp. 137–150 (In Russian)</TD></TR><TR><TD valign="top">[34]</TD> <TD valign="top"> A.A. Samarskii, ''Uspekhi Mat. Nauk'' , '''5''' : 3 (1950) pp. 133–134</TD></TR><TR><TD valign="top">[35]</TD> <TD valign="top"> A.N. Tikhonov, "Regularization of incorrectly posed problems" ''Soviet Math. Dokl.'' , '''4''' : 6 (1963) pp. 1624–1627 ''Dokl. Akad. Nauk SSSR'' , '''153''' : 1 (1963) pp. 49–52 {{MR|}} {{ZBL|0183.11601}} </TD></TR><TR><TD valign="top">[36]</TD> <TD valign="top"> I.M. Gel'fand, B.M. Levitan, "On a simple identity for the characteristic values of a second-order differential operator" ''Dokl. Akad. Nauk SSSR'' , '''88''' : 4 (1953) pp. 593–596 (In Russian)</TD></TR><TR><TD valign="top">[37]</TD> <TD valign="top"> V.B. Lidskii, V.A. Sadovnichii, "Regularized sums of zeros for a class of entire functions" ''Soviet Math. Dokl.'' , '''8''' : 5 (1967) pp. 1082–1085 ''Dokl. Akad. Nauk SSSR'' , '''176''' : 2 (1967) pp. 259–262</TD></TR><TR><TD valign="top">[38]</TD> <TD valign="top"> V.A. Sadovnichii, "The zeta-function and eigenvalues of differential operators" ''Diff. Eq.'' , '''10''' : 7 (1974) pp. 986–994 ''Differentsial. Uravnen.'' , '''10''' : 7 (1974) pp. 1276–1285 {{MR|}} {{ZBL|0318.34037}} </TD></TR><TR><TD valign="top">[39]</TD> <TD valign="top"> V.A. Lyubishkin, V.A. Sadovnichii, "Regularized traces of discrete operators" ''Soviet Math. Dokl.'' , '''24''' : 3 (1981) pp. 514–517 ''Dokl. Akad. Nauk SSSR'' , '''261''' : 2 (1981) pp. 290–293 {{MR|0638918}} {{ZBL|0496.47015}} </TD></TR><TR><TD valign="top">[40]</TD> <TD valign="top"> M.V. Keldysh, V.B. Lidskii, "On the spectral theory of nonselfadjoint operators" , ''Fourth All-Union Math. Congress'' , '''1''' , Leningrad (1963) pp. 101–120 (In Russian) {{MR|171183}} {{ZBL|}} </TD></TR><TR><TD valign="top">[41]</TD> <TD valign="top"> P.D. Lax, R.S. Philipps, "Scattering theory for automorphic functions" , Princeton Univ. Press (1976) {{MR|0562288}} {{ZBL|0362.10022}} </TD></TR><TR><TD valign="top">[42]</TD> <TD valign="top"> B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French) {{MR|275190}} {{ZBL|}} </TD></TR><TR><TD valign="top">[43]</TD> <TD valign="top"> V.P. Maslov, "Operational methods" , MIR (1976) (Translated from Russian) {{MR|0512495}} {{ZBL|0449.47002}} </TD></TR><TR><TD valign="top">[44]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , '''3''' , Interscience (1971) {{MR|0412888}} {{ZBL|}} </TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.V. Keldysh, "On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations" ''Dokl. Akad. Nauk SSSR'' , '''77''' : 1 (1951) pp. 11–14 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.V. Keldysh, "On the completeness of the eigenfunctions of some class of non-selfadjoint linear operators" ''Russian Math. Surveys'' , '''26''' : 4 (1971) pp. 15–44 ''Uspekhi Mat. Nauk'' , '''26''' : 4 (1971) pp. 15–41</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.V. Keldysh, "On a Tauberian theorem" ''Trudy Mat. Inst. Steklov.'' , '''38''' (1951) pp. 77–86 (In Russian) {{MR|}} {{ZBL|0272.40007}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D.E. Allakhverdiev, "On completeness of a system of eigenelements and associated elements of a nonselfadjoint operator close to normal" ''Dokl. Akad. Nauk SSSR'' , '''115''' : 2 (1957) pp. 207–210 (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H.K. Langer, "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728086.png" />-Hermitian operators" ''Soviet Math. Dokl.'' , '''1''' : 4 (1960) pp. 1052–1055 ''Dokl. Akad. Nauk SSSR'' , '''134''' : 2 (1960) pp. 263–266</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.G. Krein, "A new application of the fixed-point principle in the theory of operators on a space with indefinite metric" ''Soviet Math. Dokl.'' , '''5''' : 1 (1964) pp. 266–269 ''Dokl. Akad. Nauk SSSR'' , '''154''' : 5 (1964) pp. 1023–1026 {{MR|169059}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> H.K. Langer, M.G. Krein, "Applications of function theory in the mechanics of a continuous medium" , Moscow (1965) (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> L.S. Pontryagin, "Hermitian operators in spaces with indefinite metric" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''8''' (1944) pp. 243–280 (In Russian) {{MR|}} {{ZBL|0061.26004}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> V.B. Lidskii, "On completeness of a system of eigenelements and associated functions of a nonselfadjoint operator" ''Dokl. Akad. Nauk. SSSR'' , '''110''' : 2 (1956) pp. 172–175 (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M.S. Agranovich, "Summability of series in root vectors of non-self-adjoint elliptic operators" ''Funct. Anal. Appl.'' , '''10''' : 3 (1976) pp. 1–3 ''Funktsional. Anal. i Prilozhen.'' , '''10''' : 3 (1976) pp. 1–12 {{MR|}} {{ZBL|0362.35062}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , ''Transl. Math. Monogr.'' , '''18''' , Amer. Math. Soc. (1969) (Translated from Russian) {{MR|0246142}} {{ZBL|0181.13504}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> V.E. Katsnel'son, "Conditions under which systems of eigenvectors of some classes of operators form a basis" ''Funct. Anal. Appl.'' , '''1''' : 2 (1967) pp. 122–133 ''Funksional. Anal. i Prilozhen.'' , '''1''' : 2 (1967) pp. 39–51</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> B.V. Lidskii, "Conditions for complete continuity of the resolvent of a differential operator" ''Dokl. Akad. Nauk SSSR'' , '''113''' : 1 (1957) pp. 28–31 (In Russian) {{MR|0091385}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A.G. Kostyuchenko, "Asymptotic distribution of the eigenvalues of elliptic operators" ''Soviet Math. Dokl.'' , '''5''' : 5 (1964) pp. 1171–1175 ''Dokl. Akad. Nauk. SSSR'' , '''158''' : 1 (1964) pp. 41–44</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> S.M. Ponomarev, "A generalization of M.V. Keldysh's theorem concerning the completeness of eigen- and associated functions of the first boundary value problem for a nonself-adjoint elliptic operator" ''Diff. Eq.'' , '''10''' : 12 (1974) pp. 1777–1779 ''Differentsial. Uravnen.'' , '''10''' : 12 (1974) pp. 2294–2296</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> N.M. Krukovskii, "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728087.png" />-tuple completeness of the system of generalized eigenfunctions and associated functions of a non-self-adjoint elliptic operator" ''Diff. Eq.'' , '''12''' : 10 (1976) pp. 1290–1296 ''Differentsial. Uravnen.'' , '''12''' : 10 (1976) pp. 1832–1851</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Theory and applications of Volterra operators in Hilbert space" , Amer. Math. Soc. (1970) (Translated from Russian) {{MR|0264447}} {{ZBL|0194.43804}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> M.S. Brodskii, G.E. Kisilevskii, "Criterion for unicellularity of dissipative Volterra operators with nuclear imaginary components" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''30''' : 6 (1966) pp. 1213–1228 (In Russian) {{MR|0203459}} {{ZBL|}} </TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> N.K. Nikol'skii, "The unicellularity and nonunicellularity of weighted shift operators" ''Soviet Math. Dokl.'' , '''8''' : 1 (1967) pp. 91–94 ''Dokl. Akad. Nauk SSSR'' , '''172''' : 2 (1967) pp. 287–290 {{MR|}} {{ZBL|0157.21202}} </TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> M.A. Naimark, "Investigation of the spectrum and the expansion in eigenfunctions of a second-order non-self-adjoint operator on a semi-axis" ''Trudy Moskov. Mat. Obshch.'' , '''3''' (1954) pp. 181–270 (In Russian)</TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top"> M.A. Naimark, "Linear differential operators" , '''1–2''' , Harrap (1968) (Translated from Russian) {{MR|0262880}} {{ZBL|0227.34020}} </TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top"> M.G. Gasymov, "Expansion in solutions of the scattering problem for a nonselfadjoint Schrödinger equation" ''Dokl. Akad. Nauk AzSSR'' , '''22''' : 10 (1966) pp. 9–12 (In Russian)</TD></TR><TR><TD valign="top">[23]</TD> <TD valign="top"> V.A. Marchenko, "Eigenfunction expansions of non-self-adjoint singular second-order differential operators" ''Mat. Sb.'' , '''52''' (1960) pp. 739–788 (In Russian)</TD></TR><TR><TD valign="top">[24]</TD> <TD valign="top"> V.A. Sadovnichii, "Analytic properties of solutions of system of equations with singularities that form segments" ''Trudy Sem. Petrov.'' , '''2''' (1976) pp. 211–221 (In Russian)</TD></TR><TR><TD valign="top">[25]</TD> <TD valign="top"> Ya.D. Tamarkin, "On some general problems of the theory of ordinary linear differential equations..." , Petrograd (1917) (In Russian)</TD></TR><TR><TD valign="top">[26a]</TD> <TD valign="top"> L.E. Ward, "An irregular boundary value and expansion problem" ''Ann. of Math. (2)'' , '''26''' (1925) pp. 21–36 {{MR|1502673}} {{ZBL|50.0314.03}} </TD></TR><TR><TD valign="top">[26b]</TD> <TD valign="top"> L.E. Ward, "A third order irregular boundary value problem and the associated series" ''Trans. Amer. Math. Soc.'' , '''34''' (1932) pp. 417–434 {{MR|1501646}} {{ZBL|0004.11001}} {{ZBL|58.0467.02}} </TD></TR><TR><TD valign="top">[27]</TD> <TD valign="top"> A.P. Khromov, , ''Proc. 2-nd Conf. Inst. Higher Education'' , '''1''' , Kuibyshev (1962) pp. 109–113 (In Russian)</TD></TR><TR><TD valign="top">[28]</TD> <TD valign="top"> V.P. Mikhailov, "Riesz bases in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728088.png" />" ''Soviet Math. Dokl.'' , '''3''' : 3 (1962) pp. 851–854 ''Dokl. Akad. Nauk SSSR'' , '''144''' : 5 (1962) pp. 981–984</TD></TR><TR><TD valign="top">[29]</TD> <TD valign="top"> G.M. Kessel'man, ''Izv. Vyzov. Mat.'' , '''2''' (1964) pp. 82–93</TD></TR><TR><TD valign="top">[30]</TD> <TD valign="top"> V.A. Il'in, "On the properties of a reduced system of eigenfunctions and associated functions of a Keldysh bundle of ordinary differential operators" ''Soviet Math. Dokl.'' , '''17''' : 5 (1976) pp. 1247–1250 ''Dokl. Akad. Nauk SSSR'' , '''230''' (1976) pp. 30–33</TD></TR><TR><TD valign="top">[31]</TD> <TD valign="top"> E.N. Moisev, "The uniform convergence of certain expansions in a closed domain" ''Soviet Math. Dokl.'' , '''18''' : 2 (1977) pp. 549–553 ''Dokl. Akad. Nauk SSSR'' , '''233''' (1977) pp. 1042–1045</TD></TR><TR><TD valign="top">[32]</TD> <TD valign="top"> V.V. Tikhomirov, "On the Riesz means of expansions in eigenfunctions and associated functions of a nonselfadjoint ordinary differential operator" ''Math. USSR Sb.'' , '''31''' : 1 (1977) pp. 29–48 ''Mat. Sb.'' , '''102''' : 1 (1977) pp. 33–55 {{MR|}} {{ZBL|0388.47027}} </TD></TR><TR><TD valign="top">[33]</TD> <TD valign="top"> V.B. Lidskii, "Fourier series expansion of principal functions of a non-self-adjoint elliptic operator" ''Mat. Sb.'' , '''57''' : 2 (1962) pp. 137–150 (In Russian)</TD></TR><TR><TD valign="top">[34]</TD> <TD valign="top"> A.A. Samarskii, ''Uspekhi Mat. Nauk'' , '''5''' : 3 (1950) pp. 133–134</TD></TR><TR><TD valign="top">[35]</TD> <TD valign="top"> A.N. Tikhonov, "Regularization of incorrectly posed problems" ''Soviet Math. Dokl.'' , '''4''' : 6 (1963) pp. 1624–1627 ''Dokl. Akad. Nauk SSSR'' , '''153''' : 1 (1963) pp. 49–52 {{MR|}} {{ZBL|0183.11601}} </TD></TR><TR><TD valign="top">[36]</TD> <TD valign="top"> I.M. Gel'fand, B.M. Levitan, "On a simple identity for the characteristic values of a second-order differential operator" ''Dokl. Akad. Nauk SSSR'' , '''88''' : 4 (1953) pp. 593–596 (In Russian)</TD></TR><TR><TD valign="top">[37]</TD> <TD valign="top"> V.B. Lidskii, V.A. Sadovnichii, "Regularized sums of zeros for a class of entire functions" ''Soviet Math. Dokl.'' , '''8''' : 5 (1967) pp. 1082–1085 ''Dokl. Akad. Nauk SSSR'' , '''176''' : 2 (1967) pp. 259–262</TD></TR><TR><TD valign="top">[38]</TD> <TD valign="top"> V.A. Sadovnichii, "The zeta-function and eigenvalues of differential operators" ''Diff. Eq.'' , '''10''' : 7 (1974) pp. 986–994 ''Differentsial. Uravnen.'' , '''10''' : 7 (1974) pp. 1276–1285 {{MR|}} {{ZBL|0318.34037}} </TD></TR><TR><TD valign="top">[39]</TD> <TD valign="top"> V.A. Lyubishkin, V.A. Sadovnichii, "Regularized traces of discrete operators" ''Soviet Math. Dokl.'' , '''24''' : 3 (1981) pp. 514–517 ''Dokl. Akad. Nauk SSSR'' , '''261''' : 2 (1981) pp. 290–293 {{MR|0638918}} {{ZBL|0496.47015}} </TD></TR><TR><TD valign="top">[40]</TD> <TD valign="top"> M.V. Keldysh, V.B. Lidskii, "On the spectral theory of nonselfadjoint operators" , ''Fourth All-Union Math. Congress'' , '''1''' , Leningrad (1963) pp. 101–120 (In Russian) {{MR|171183}} {{ZBL|}} </TD></TR><TR><TD valign="top">[41]</TD> <TD valign="top"> P.D. Lax, R.S. Philipps, "Scattering theory for automorphic functions" , Princeton Univ. Press (1976) {{MR|0562288}} {{ZBL|0362.10022}} </TD></TR><TR><TD valign="top">[42]</TD> <TD valign="top"> B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French) {{MR|275190}} {{ZBL|}} </TD></TR><TR><TD valign="top">[43]</TD> <TD valign="top"> V.P. Maslov, "Operational methods" , MIR (1976) (Translated from Russian) {{MR|0512495}} {{ZBL|0449.47002}} </TD></TR><TR><TD valign="top">[44]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , '''3''' , Interscience (1971) {{MR|0412888}} {{ZBL|}} </TD></TR></table> | ||
+ | |||
+ | |||
====Comments==== | ====Comments==== | ||
The main article above gives a partial picture of the theory of non-self-adjoint operators. The method of Keldysh about multiple completeness is explained in detail. A second important method, which has been developed by M.G. Krein and H. Langer (see [[#References|[7]]]), is only touched upon. The Krein–Langer method uses factorization as a tool. | The main article above gives a partial picture of the theory of non-self-adjoint operators. The method of Keldysh about multiple completeness is explained in detail. A second important method, which has been developed by M.G. Krein and H. Langer (see [[#References|[7]]]), is only touched upon. The Krein–Langer method uses factorization as a tool. | ||
− | Consider the quadratic operator polynomial | + | Consider the quadratic operator polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728089.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728091.png" /> are bounded linear operators acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728092.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728093.png" /> be a (right) operator root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728094.png" />, that is, |
− | where | ||
− | and | ||
− | are bounded linear operators acting on a Hilbert space | ||
− | and let | ||
− | be a (right) operator root of | ||
− | that is, | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728095.png" /></td> </tr></table> | |
− | |||
− | |||
− | Then | + | Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728096.png" /> is a solution of the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728097.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728098.png" />. One of the problems is to find operator roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n06728099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280100.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280101.png" /> so that |
− | is a solution of the differential equation | ||
− | for each | ||
− | One of the problems is to find operator roots | ||
− | and | ||
− | of | ||
− | so that | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280102.png" /></td> </tr></table> | |
− | |||
− | |||
− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280104.png" /> run over all vectors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280105.png" />, form the set of all solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280106.png" />. Moreover, if such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280108.png" /> have been found, one would like to know when the linear span of elementary solutions of the equations |
− | and | ||
− | run over all vectors in | ||
− | form the set of all solutions of | ||
− | Moreover, if such | ||
− | and | ||
− | have been found, one would like to know when the linear span of elementary solutions of the equations | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280109.png" /></td> </tr></table> | |
− | |||
− | |||
− | is dense (in a sense to be made more precise) in the space of all solutions. If | + | is dense (in a sense to be made more precise) in the space of all solutions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280110.png" /> is an operator root, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280111.png" /> is a right divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280112.png" />, that is, |
− | is an operator root, then | ||
− | is a right divisor of | ||
− | that is, | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280113.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table> | |
− | |||
− | |||
− | |||
− | The method of Krein–Langer uses the theory of spaces with an indefinite metric (cf. [[Space with an indefinite metric|Space with an indefinite metric]]) to obtain factorizations of the type (a1) and to analyze their properties for the important case when | + | The method of Krein–Langer uses the theory of spaces with an indefinite metric (cf. [[Space with an indefinite metric|Space with an indefinite metric]]) to obtain factorizations of the type (a1) and to analyze their properties for the important case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280115.png" /> are self-adjoint. Here indefinite metrics turn up naturally, because the companion operator |
− | and | ||
− | are self-adjoint. Here indefinite metrics turn up naturally, because the companion operator | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280116.png" /></td> </tr></table> | |
− | |||
is self-adjoint relative to the indefinite metric | is self-adjoint relative to the indefinite metric | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280117.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | < | ||
− | |||
− | whenever | + | whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280119.png" /> are self-adjoint relative to the usual product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280120.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280121.png" />. The Krein–Langer method is, therefore, of a geometric nature, while Keldysh's method has an analytic character. |
− | and | ||
− | are self-adjoint relative to the usual product | ||
− | on | ||
− | The Krein–Langer method is, therefore, of a geometric nature, while Keldysh's method has an analytic character. | ||
The Krein–Langer approach has led to two directions of research. One continued to use and develop further the geometric methods of operator theory in spaces with an indefinite metric, with Langer as one of the main contributors. The other direction adapted Wiener–Hopf factorization methods for operator-valued functions and used these analytical tools to study the spectral properties of polynomials and analytic operator functions. Here the main contributors are A.S. Markus and V.I. Matsaev. The Keldysh method and the Krein–Langer method are nicely presented in [[#References|[a1]]]. Parallel, but with a stronger emphasis on the finite-dimensional case, the factorization direction was also developed in the Western literature. In the latter, connections with mathematical systems theory played an important role (see, e.g., [[#References|[a2]]]–[[#References|[a4]]]). | The Krein–Langer approach has led to two directions of research. One continued to use and develop further the geometric methods of operator theory in spaces with an indefinite metric, with Langer as one of the main contributors. The other direction adapted Wiener–Hopf factorization methods for operator-valued functions and used these analytical tools to study the spectral properties of polynomials and analytic operator functions. Here the main contributors are A.S. Markus and V.I. Matsaev. The Keldysh method and the Krein–Langer method are nicely presented in [[#References|[a1]]]. Parallel, but with a stronger emphasis on the finite-dimensional case, the factorization direction was also developed in the Western literature. In the latter, connections with mathematical systems theory played an important role (see, e.g., [[#References|[a2]]]–[[#References|[a4]]]). | ||
− | A third major method in the theory of non-self-adjoint operators, probably the oldest in this area, was initiated in the middle of the 1940s by M.S. Lifshits. The main tool here is the characteristic operator function, which is associated with an arbitrary operator | + | A third major method in the theory of non-self-adjoint operators, probably the oldest in this area, was initiated in the middle of the 1940s by M.S. Lifshits. The main tool here is the characteristic operator function, which is associated with an arbitrary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280122.png" /> in the following way: |
− | in the following way: | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280123.png" /></td> </tr></table> | |
− | |||
− | |||
− | where | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280124.png" /> is a signature operator (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280125.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280126.png" />. This function serves as a unitary invariant for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280127.png" /> (some trivial self-adjoint parts excluded), and in several important cases it is much easier to analyze than the original operator. The characteristic operator function has intriguing properties, for example, from its divisors invariant subspaces of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280128.png" /> may be read off. The theory of characteristic operator functions has been extended in many different directions by M.S. Brodskii, L.A. Sakhnovich, I.C. Gohberg and Krein (see [[#References|[a5]]], [[#References|[a6]]], [[#References|[11]]], [[#References|[17]]]). A far-reaching development connected with the theory of characteristic operator functions concerns the study of contractive operators by B. Sz.-Nagy and C. Foiaş [[#References|[42]]], which has led to a complete understanding of a wide class of non-self-adjoint operators [[#References|[a7]]]. |
− | is a signature operator (i.e. | ||
− | and | ||
− | This function serves as a unitary invariant for the operator | ||
− | some trivial self-adjoint parts excluded), and in several important cases it is much easier to analyze than the original operator. The characteristic operator function has intriguing properties, for example, from its divisors invariant subspaces of the operator | ||
− | may be read off. The theory of characteristic operator functions has been extended in many different directions by M.S. Brodskii, L.A. Sakhnovich, I.C. Gohberg and Krein (see [[#References|[a5]]], [[#References|[a6]]], [[#References|[11]]], [[#References|[17]]]). A far-reaching development connected with the theory of characteristic operator functions concerns the study of contractive operators by B. Sz.-Nagy and C. Foiaş [[#References|[42]]], which has led to a complete understanding of a wide class of non-self-adjoint operators [[#References|[a7]]]. | ||
Another original method in the theory of non-self-adjoint operators is due to L. de Branges and his co-workers ([[#References|[a8]]], [[#References|[a9]]]). The method of de Branges is based on a deep analysis of the theory of spaces of entire vector-valued functions. Here the mapping | Another original method in the theory of non-self-adjoint operators is due to L. de Branges and his co-workers ([[#References|[a8]]], [[#References|[a9]]]). The method of de Branges is based on a deep analysis of the theory of spaces of entire vector-valued functions. Here the mapping | ||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067280/n067280129.png" /></td> </tr></table> | |
− | |||
− | |||
provides the main model. Recently this approach led to a sensational success by yielding a proof of the [[Bieberbach conjecture|Bieberbach conjecture]] [[#References|[a10]]]. | provides the main model. Recently this approach led to a sensational success by yielding a proof of the [[Bieberbach conjecture|Bieberbach conjecture]] [[#References|[a10]]]. |
Revision as of 14:52, 7 June 2020
A linear operator in a Hilbert space the spectral analysis of which cannot be made to fit into the framework of the theory of self-adjoint operators (cf. Self-adjoint operator) and its simplest generalizations: the theory of unitary operators (cf. Unitary operator) and the theory of normal operators (cf. Normal operator). Non-self-adjoint operators arise in the discussion of processes that proceed without conservation of energy: in problems with friction, in the theory of open resonators, in problems of inelastic scattering, and others. Certain self-adjoint problems, in which by separation of variables an operator-valued function appears that depends non-linearly on a spectral parameter
, also lead to a study of non-self-adjoint operators. Many of the propositions referring to the theory of non-self-adjoint operators are valid also for operators acting in arbitrary Banach spaces,
-spaces, topological vector spaces, etc.
The most extensive method in the study of non-self-adjoint operators is that of evaluating the resolvent, which makes use of the theory of analytic functions, of asymptotic expansions, etc.
The first works concerned with the theory of non-self-adjoint operators were by G. Birkhoff, Ya.D. Tamarkin, V.A. Steklov, and others, in the investigation of problems for ordinary differential equations. These studies applied Cauchy's method of contour integration to the resolvent.
For non-self-adjoint partial differential operators effective methods of research were lacking for a long time. This can be explained by the complicated structure of the resolvent of such an operator as an analytic function.
In the development of a general theory of non-self-adjoint operators (in particular, partial differential operators) an important role was played by the work of M.V. Keldysh [1] (see also [2]). He studied an equation of the form
![]() | (1) |
where is an element of a certain Hilbert space
and the operator
has the representation
![]() |
Here is a completely-continuous invertible self-adjoint operator of finite order and the
,
, are arbitrary completely-continuous operators. (A completely-continuous operator
acting on a Hilbert space is said to be an operator of finite order if
for some
,
; the
denote the singular numbers of
, that is, the eigen values of
.) The eigen values of (1) are those
for which the equation has non-trivial solutions
; these solutions are called eigen vectors.
Under the assumptions made above the spectrum of (1) is discrete. Since is non-self-adjoint, apart from the eigen vectors there naturally arise (in the presence of a multiple spectrum) associated vectors. In [1] a chain is constructed of associated vectors
corresponding to an eigen value
and eigen vector
according to the rule
![]() | (2) |
![]() |
The system of eigen vectors and associated vectors of is said to be
-fold complete if any
vectors
of
can be approximated in the norm of
with arbitrary accuracy by finite linear combinations of the form
![]() |
with the same coefficients . Here
a is vector-valued function of the form
![]() |
The definition of -fold completeness is, naturally, connected with the solution of the Cauchy problem for the non-stationary equation corresponding to (1).
According to a theorem of Keldysh, under the assumptions made on the coefficients of the system of all eigen vectors and associated vectors of
is
-fold complete in
. In [1] he also proved that the eigen values of
can be approximated asymptotically on rays with
. In the proof of completeness Keldysh developed a new method for evaluating the resolvent of an abstract completely-continuous non-self-adjoint operator of finite order. Here emerged the special role played in the completeness problem by Volterra operators, i.e. completely-continuous operators with a single point of the spectrum at zero. In establishing the asymptotic behaviour of the eigen values, Keldysh [3] used a new Tauberian theorem due to him.
Keldysh's research was continued by many authors. His theorem was extended in [4] to the case when the operator depends rationally on
.
In [5]–[7] the equation with
is considered, where
is a completely-continuous positive-definite operator and
is a bounded self-adjoint operator. A generalization of Pontryagin's theorem (see [8]) on the existence of a maximal
-non-negative invariant subspace for a
-self-adjoint operator
made it possible (see [6] and [7]) to establish the two-fold completeness of all eigen vectors and associated vectors of
in situations important for applications, and also the one-fold completeness of the subsystem corresponding to the spectrum situated in the left (or right) half-plane. These results have been developed much further.
The summability of Fourier series in eigen vectors and associated vectors of a completely-continuous operator of finite order
has been established (see [9]) if the eigen values of the quadratic form
lie in a sector of the complex plane with opening angle less than
. (For applications of this theorem and further generalizations of it see [10] and the references there.)
The question when the system of eigen vectors and associated vectors forms a basis in the Hilbert space has been studied in a number of papers. Most general conditions under which the system of eigen vectors and associated vectors of a dissipative completely-continuous operator forms a basis are found in [12].
In the case of singular differential operators with discrete spectrum a number of subtle results have been obtained (see [11] and [13]) on the completeness of the eigen functions and associated functions of a Sturm–Liouville operator with complex potential. Important results have been obtained for the case of an elliptic operator (see [14]). Keldysh's theorem has been generalized to the case of generalized eigen functions and associated functions of non-self-adjoint elliptic operators (see [15], [16]).
An attempt to carry over the theorem on the reduction to Jordan form of a finite-dimensional operator to the infinite-dimensional case has led to the construction of a triangular integral representation. For completely-continuous operators , where
and
are self-adjoint and
is of finite order, an analogue has been obtained of Schur's theorem on the unitary equivalence of
to a triangular operator (see [17]). A special place in the problem of triangular representation is occupied by Volterra operators.
A special role in this problem is played by von Neumann's theorem, stating that a completely-continuous linear operator in a Hilbert space has a non-trivial invariant subspace; an arbitrary bounded linear operator in a Banach space need not have an invariant subspace; the corresponding problem for the case of a Hilbert space is still open (1989). A Volterra operator is said to be a unicellular operator if for any two invariant subspaces and
of it either
or
. In [18] a necessary and sufficient condition was found for an operator
to be a unicellular operator under the assumption that
is nuclear and non-negative definite; this condition can be stated in terms of the growth of the resolvent of
as
. A simple sufficient condition for being a unicellular operator is indicated in [19].
Non-self-adjoint operators with a continuous spectrum were first studied by M.A. Naimark (see [20], [21]), who obtained an expansion in a Fourier integral, connected with the non-self-adjoint problem
![]() | (3) |
![]() | (4) |
where is a complex-valued function subject to the condition
![]() | (5) |
and is a complex number. The results in [20] imply, in particular, that in a neighbourhood of points of the real axis where
vanishes
, the spectral projections of the operator (3)–(4) are unbounded. (Here
denotes the solution of (3) for which
as
, the Jost solution.) In [20], the real zeros of
are called spectral singularities. In [22] a generalization of the results of [20] to the case of the Schrödinger equation in the three-dimensional space is obtained. In a further development of [20] it has been proved (see [23]) that in general (without a restriction of type (5)) the spectral function of a differential operator must be regarded as a continuous linear functional on a certain topological space.
In [24] a system of non-self-adjoint equations is studied with singular points whose position depends on the spectral parameter. These systems occur in the theory of shells without moments (cf. Shell theory). For such systems asymptotic properties of the solutions have been established, and also solvability theorems of Cauchy type have been proved. Completeness theorems for the system of eigen functions and associated functions of non-self-adjoint integro-differential operators generating a non-regular problem have been established.
An important problem in the theory of non-self-adjoint operators is the expansion of the kernel of the Green operator of a biorthogonal series in eigen functions and associated functions and also the question of a basis. Tamarkin [25] has studied the expansion of a summable function in a series of eigen functions and associated functions of a regular problem, and also the question of equiconvergence with a trigonometric Fourier series. Later it was proved (see , [27]) that for non-regular problems equiconvergence with a trigonometric series does not hold.
For strongly-regular conditions the system of eigen functions and associated functions forms a basis in . It has been proved (see [28], [29]) that this system not only forms a basis, but even a so-called Riesz basis.
In [30] an important method has been developed for studying the basis property and the uniform convergence of the expansions in eigen functions and associated functions of an ordinary non-self-adjoint operator. This method is a further development of the ideas applied in the investigation of self-adjoint problems. A new treatment for the eigen functions and associated functions has been proposed, making it possible to dispense with the specific form of the boundary conditions; general ordinary differential operators or pencils of such operators are considered and necessary and sufficient conditions have been established for the basis property of the eigen functions and associated functions of such operators, as well as a criterion for equiconvergence. The method is based only on a mean value formula for the eigen functions and associated functions (see also [31]). It also turns out that if the operator has infinitely many associated functions, then the basis property depends on the choice of the latter (see [32]).
For non-self-adjoint elliptic operators there is convergence (see [33]) of a certain sequence of means of Poisson type of the partial sums of the biorthogonal series, that is, a summation method has been proposed.
Expansions in eigen functions and associated functions of non-regular problems were first obtained for problems of the form ,
. It has been proved (see ) that expansions in uniformly convergent series of this form can hold for functions satisfying a certain analyticity condition.
Among the fundamental studies is the problem (see [34]) in which the perturbation of the spectrum of the Laplace operator under a change of domain is studied. Here for the first time the role of the capacity of the set which is varying on the spectrum of the operator became manifest. These methods were successfully applied in the study of non-self-adjoint operators.
Regularization methods, the basics of which were expounded in [35], are successfully used in the theory of non-self-adjoint operators. An example is the problem of regularized traces (cf. Trace) of non-self-adjoint operators. The first paper on the theory of traces was [36], where the regularized trace of a Sturm–Liouville operator was computed. More general results in the theory of operator traces were obtained in [37] and [38]. It turns out that trace formulas for ordinary non-self-adjoint differential operators depending in a complicated manner on the spectral parameter can be obtained as consequences of the formulas for regularized sums of roots of a certain class of entire functions. (The traces of singular operators and partial differential operators are considered in [38], [39].)
Among the important works in which new methods and ideas in the theory of non-self-adjoint operators were developed one counts also the survey lecture [40].
The construction of a definitive theory of non-self-adjoint operators is far from complete (1989). On the one hand, in the theory itself there are new trends of research, such as scattering theory [41], the construction of the theory of contraction operators [42], the method of the canonical Maslov operator [43], the theory of spectral operators [44], and others; on the other hand, research on applied problems, mechanics and mathematical physics suggests new ways of developing this theory.
References
[1] | M.V. Keldysh, "On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations" Dokl. Akad. Nauk SSSR , 77 : 1 (1951) pp. 11–14 (In Russian) |
[2] | M.V. Keldysh, "On the completeness of the eigenfunctions of some class of non-selfadjoint linear operators" Russian Math. Surveys , 26 : 4 (1971) pp. 15–44 Uspekhi Mat. Nauk , 26 : 4 (1971) pp. 15–41 |
[3] | M.V. Keldysh, "On a Tauberian theorem" Trudy Mat. Inst. Steklov. , 38 (1951) pp. 77–86 (In Russian) Zbl 0272.40007 |
[4] | D.E. Allakhverdiev, "On completeness of a system of eigenelements and associated elements of a nonselfadjoint operator close to normal" Dokl. Akad. Nauk SSSR , 115 : 2 (1957) pp. 207–210 (In Russian) |
[5] | H.K. Langer, "On ![]() |
[6] | M.G. Krein, "A new application of the fixed-point principle in the theory of operators on a space with indefinite metric" Soviet Math. Dokl. , 5 : 1 (1964) pp. 266–269 Dokl. Akad. Nauk SSSR , 154 : 5 (1964) pp. 1023–1026 MR169059 |
[7] | H.K. Langer, M.G. Krein, "Applications of function theory in the mechanics of a continuous medium" , Moscow (1965) (In Russian) |
[8] | L.S. Pontryagin, "Hermitian operators in spaces with indefinite metric" Izv. Akad. Nauk SSSR Ser. Mat. , 8 (1944) pp. 243–280 (In Russian) Zbl 0061.26004 |
[9] | V.B. Lidskii, "On completeness of a system of eigenelements and associated functions of a nonselfadjoint operator" Dokl. Akad. Nauk. SSSR , 110 : 2 (1956) pp. 172–175 (In Russian) |
[10] | M.S. Agranovich, "Summability of series in root vectors of non-self-adjoint elliptic operators" Funct. Anal. Appl. , 10 : 3 (1976) pp. 1–3 Funktsional. Anal. i Prilozhen. , 10 : 3 (1976) pp. 1–12 Zbl 0362.35062 |
[11] | I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , Transl. Math. Monogr. , 18 , Amer. Math. Soc. (1969) (Translated from Russian) MR0246142 Zbl 0181.13504 |
[12] | V.E. Katsnel'son, "Conditions under which systems of eigenvectors of some classes of operators form a basis" Funct. Anal. Appl. , 1 : 2 (1967) pp. 122–133 Funksional. Anal. i Prilozhen. , 1 : 2 (1967) pp. 39–51 |
[13] | B.V. Lidskii, "Conditions for complete continuity of the resolvent of a differential operator" Dokl. Akad. Nauk SSSR , 113 : 1 (1957) pp. 28–31 (In Russian) MR0091385 |
[14] | A.G. Kostyuchenko, "Asymptotic distribution of the eigenvalues of elliptic operators" Soviet Math. Dokl. , 5 : 5 (1964) pp. 1171–1175 Dokl. Akad. Nauk. SSSR , 158 : 1 (1964) pp. 41–44 |
[15] | S.M. Ponomarev, "A generalization of M.V. Keldysh's theorem concerning the completeness of eigen- and associated functions of the first boundary value problem for a nonself-adjoint elliptic operator" Diff. Eq. , 10 : 12 (1974) pp. 1777–1779 Differentsial. Uravnen. , 10 : 12 (1974) pp. 2294–2296 |
[16] | N.M. Krukovskii, "The ![]() |
[17] | I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Theory and applications of Volterra operators in Hilbert space" , Amer. Math. Soc. (1970) (Translated from Russian) MR0264447 Zbl 0194.43804 |
[18] | M.S. Brodskii, G.E. Kisilevskii, "Criterion for unicellularity of dissipative Volterra operators with nuclear imaginary components" Izv. Akad. Nauk SSSR Ser. Mat. , 30 : 6 (1966) pp. 1213–1228 (In Russian) MR0203459 |
[19] | N.K. Nikol'skii, "The unicellularity and nonunicellularity of weighted shift operators" Soviet Math. Dokl. , 8 : 1 (1967) pp. 91–94 Dokl. Akad. Nauk SSSR , 172 : 2 (1967) pp. 287–290 Zbl 0157.21202 |
[20] | M.A. Naimark, "Investigation of the spectrum and the expansion in eigenfunctions of a second-order non-self-adjoint operator on a semi-axis" Trudy Moskov. Mat. Obshch. , 3 (1954) pp. 181–270 (In Russian) |
[21] | M.A. Naimark, "Linear differential operators" , 1–2 , Harrap (1968) (Translated from Russian) MR0262880 Zbl 0227.34020 |
[22] | M.G. Gasymov, "Expansion in solutions of the scattering problem for a nonselfadjoint Schrödinger equation" Dokl. Akad. Nauk AzSSR , 22 : 10 (1966) pp. 9–12 (In Russian) |
[23] | V.A. Marchenko, "Eigenfunction expansions of non-self-adjoint singular second-order differential operators" Mat. Sb. , 52 (1960) pp. 739–788 (In Russian) |
[24] | V.A. Sadovnichii, "Analytic properties of solutions of system of equations with singularities that form segments" Trudy Sem. Petrov. , 2 (1976) pp. 211–221 (In Russian) |
[25] | Ya.D. Tamarkin, "On some general problems of the theory of ordinary linear differential equations..." , Petrograd (1917) (In Russian) |
[26a] | L.E. Ward, "An irregular boundary value and expansion problem" Ann. of Math. (2) , 26 (1925) pp. 21–36 MR1502673 Zbl 50.0314.03 |
[26b] | L.E. Ward, "A third order irregular boundary value problem and the associated series" Trans. Amer. Math. Soc. , 34 (1932) pp. 417–434 MR1501646 Zbl 0004.11001 Zbl 58.0467.02 |
[27] | A.P. Khromov, , Proc. 2-nd Conf. Inst. Higher Education , 1 , Kuibyshev (1962) pp. 109–113 (In Russian) |
[28] | V.P. Mikhailov, "Riesz bases in ![]() |
[29] | G.M. Kessel'man, Izv. Vyzov. Mat. , 2 (1964) pp. 82–93 |
[30] | V.A. Il'in, "On the properties of a reduced system of eigenfunctions and associated functions of a Keldysh bundle of ordinary differential operators" Soviet Math. Dokl. , 17 : 5 (1976) pp. 1247–1250 Dokl. Akad. Nauk SSSR , 230 (1976) pp. 30–33 |
[31] | E.N. Moisev, "The uniform convergence of certain expansions in a closed domain" Soviet Math. Dokl. , 18 : 2 (1977) pp. 549–553 Dokl. Akad. Nauk SSSR , 233 (1977) pp. 1042–1045 |
[32] | V.V. Tikhomirov, "On the Riesz means of expansions in eigenfunctions and associated functions of a nonselfadjoint ordinary differential operator" Math. USSR Sb. , 31 : 1 (1977) pp. 29–48 Mat. Sb. , 102 : 1 (1977) pp. 33–55 Zbl 0388.47027 |
[33] | V.B. Lidskii, "Fourier series expansion of principal functions of a non-self-adjoint elliptic operator" Mat. Sb. , 57 : 2 (1962) pp. 137–150 (In Russian) |
[34] | A.A. Samarskii, Uspekhi Mat. Nauk , 5 : 3 (1950) pp. 133–134 |
[35] | A.N. Tikhonov, "Regularization of incorrectly posed problems" Soviet Math. Dokl. , 4 : 6 (1963) pp. 1624–1627 Dokl. Akad. Nauk SSSR , 153 : 1 (1963) pp. 49–52 Zbl 0183.11601 |
[36] | I.M. Gel'fand, B.M. Levitan, "On a simple identity for the characteristic values of a second-order differential operator" Dokl. Akad. Nauk SSSR , 88 : 4 (1953) pp. 593–596 (In Russian) |
[37] | V.B. Lidskii, V.A. Sadovnichii, "Regularized sums of zeros for a class of entire functions" Soviet Math. Dokl. , 8 : 5 (1967) pp. 1082–1085 Dokl. Akad. Nauk SSSR , 176 : 2 (1967) pp. 259–262 |
[38] | V.A. Sadovnichii, "The zeta-function and eigenvalues of differential operators" Diff. Eq. , 10 : 7 (1974) pp. 986–994 Differentsial. Uravnen. , 10 : 7 (1974) pp. 1276–1285 Zbl 0318.34037 |
[39] | V.A. Lyubishkin, V.A. Sadovnichii, "Regularized traces of discrete operators" Soviet Math. Dokl. , 24 : 3 (1981) pp. 514–517 Dokl. Akad. Nauk SSSR , 261 : 2 (1981) pp. 290–293 MR0638918 Zbl 0496.47015 |
[40] | M.V. Keldysh, V.B. Lidskii, "On the spectral theory of nonselfadjoint operators" , Fourth All-Union Math. Congress , 1 , Leningrad (1963) pp. 101–120 (In Russian) MR171183 |
[41] | P.D. Lax, R.S. Philipps, "Scattering theory for automorphic functions" , Princeton Univ. Press (1976) MR0562288 Zbl 0362.10022 |
[42] | B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French) MR275190 |
[43] | V.P. Maslov, "Operational methods" , MIR (1976) (Translated from Russian) MR0512495 Zbl 0449.47002 |
[44] | N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , 3 , Interscience (1971) MR0412888 |
Comments
The main article above gives a partial picture of the theory of non-self-adjoint operators. The method of Keldysh about multiple completeness is explained in detail. A second important method, which has been developed by M.G. Krein and H. Langer (see [7]), is only touched upon. The Krein–Langer method uses factorization as a tool.
Consider the quadratic operator polynomial , where
and
are bounded linear operators acting on a Hilbert space
, and let
be a (right) operator root of
, that is,
![]() |
Then is a solution of the differential equation
for each
. One of the problems is to find operator roots
and
of
so that
![]() |
where and
run over all vectors in
, form the set of all solutions of
. Moreover, if such
and
have been found, one would like to know when the linear span of elementary solutions of the equations
![]() |
is dense (in a sense to be made more precise) in the space of all solutions. If is an operator root, then
is a right divisor of
, that is,
![]() | (a1) |
The method of Krein–Langer uses the theory of spaces with an indefinite metric (cf. Space with an indefinite metric) to obtain factorizations of the type (a1) and to analyze their properties for the important case when and
are self-adjoint. Here indefinite metrics turn up naturally, because the companion operator
![]() |
is self-adjoint relative to the indefinite metric
![]() |
whenever and
are self-adjoint relative to the usual product
on
. The Krein–Langer method is, therefore, of a geometric nature, while Keldysh's method has an analytic character.
The Krein–Langer approach has led to two directions of research. One continued to use and develop further the geometric methods of operator theory in spaces with an indefinite metric, with Langer as one of the main contributors. The other direction adapted Wiener–Hopf factorization methods for operator-valued functions and used these analytical tools to study the spectral properties of polynomials and analytic operator functions. Here the main contributors are A.S. Markus and V.I. Matsaev. The Keldysh method and the Krein–Langer method are nicely presented in [a1]. Parallel, but with a stronger emphasis on the finite-dimensional case, the factorization direction was also developed in the Western literature. In the latter, connections with mathematical systems theory played an important role (see, e.g., [a2]–[a4]).
A third major method in the theory of non-self-adjoint operators, probably the oldest in this area, was initiated in the middle of the 1940s by M.S. Lifshits. The main tool here is the characteristic operator function, which is associated with an arbitrary operator in the following way:
![]() |
where is a signature operator (i.e.
) and
. This function serves as a unitary invariant for the operator
(some trivial self-adjoint parts excluded), and in several important cases it is much easier to analyze than the original operator. The characteristic operator function has intriguing properties, for example, from its divisors invariant subspaces of the operator
may be read off. The theory of characteristic operator functions has been extended in many different directions by M.S. Brodskii, L.A. Sakhnovich, I.C. Gohberg and Krein (see [a5], [a6], [11], [17]). A far-reaching development connected with the theory of characteristic operator functions concerns the study of contractive operators by B. Sz.-Nagy and C. Foiaş [42], which has led to a complete understanding of a wide class of non-self-adjoint operators [a7].
Another original method in the theory of non-self-adjoint operators is due to L. de Branges and his co-workers ([a8], [a9]). The method of de Branges is based on a deep analysis of the theory of spaces of entire vector-valued functions. Here the mapping
![]() |
provides the main model. Recently this approach led to a sensational success by yielding a proof of the Bieberbach conjecture [a10].
References
[a1] | A.S. Markus, "Introduction to the spectral theory of polynomial operator pencils" , Amer. Math. Soc. (1986) (Translated from Russian) MR0861412 Zbl 0678.47006 |
[a2] | I. Gohberg, P. Lancaster, L. Rodman, "Matrix polynomials" , Acad. Press (1982) MR0678825 MR0662418 MR0649765 Zbl 0496.15014 Zbl 0486.15008 Zbl 0482.15001 |
[a3] | H. Bart, I. Gohberg, M.A. Kaashoek, "Minimal factorization of matrix and operation functions" , Birkhäuser (1979) |
[a4] | L. Rodman, "An introduction to operator polynomials" , Birkhäuser (1989) MR0997092 Zbl 0685.47011 |
[a5] | M.S. [M.S. Lifshits] Livšic, "Operators, oscillations, waves" , Amer. Math. Soc. (1973) (Translated from Russian) MR0347396 Zbl 0254.47001 |
[a6] | M.S. Brodskii, "Triangular and Jordan representations of linear operators" , Amer. Math. Soc. (1971) (Translated from Russian) MR0322542 |
[a7] | H. Bercovici, "Operator theory and arithmetic in ![]() |
[a8] | L. de Branges, "Hilbert spaces of entire functions" , Prentice-Hall (1968) Zbl 0157.43301 |
[a9] | L. de Branges, J. Rovnyak, "Square summable power series" , Rinehart & Winston (1966) MR215065 Zbl 0153.39603 |
[a10] | L. de Branges, "A proof of the Bieberbach conjecture" Acta Math. , 154 (1985) pp. 137–152 Zbl 0573.30014 |
[a11] | M.S. Brodskii, M.S. [M.S. Lifshits] Livšic, "Spectral analysis of non-selfadjoint operators and intermediate systems" Transl. Amer. Math. Soc. (2) , 13 (1960) pp. 265–346 MR0113144 |
[a12] | H.R. Dowson, "Spectral theory of linear operators" , Acad. Press (1978) MR0511427 Zbl 0384.47001 |
Non-self-adjoint operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-self-adjoint_operator&oldid=49336