View source for Pontryagin space

A [[Hilbert space with an indefinite metric|Hilbert space with an indefinite metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738001.png" /> that has a finite rank of indefiniteness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738002.png" />. Basic facts concerning the geometry of these spaces were established by L.S. Pontryagin [[#References|[1]]]. Besides the facts common for spaces with an indefinite metric, the following properties hold.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738003.png" /> is an arbitrary non-negative linear manifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738004.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738005.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738006.png" /> is a positive linear manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738007.png" />, then its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738008.png" />-orthogonal complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p0738009.png" /> is a negative linear manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380010.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380011.png" /> is a complete space with respect to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380012.png" />. If the linear manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380013.png" /> is non-degenerate, then its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380014.png" />-orthogonal complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380015.png" /> is non-degenerate as well and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380016.png" />.

The spectrum (in particular, the discrete spectrum) of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380017.png" />-unitary (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380018.png" />-self-adjoint) operator is symmetric with respect to the unit circle (real line), all elementary divisors corresponding to eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380020.png" />, are of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380023.png" />. The sum of the dimensions of the root subspaces of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380024.png" />-unitary (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380025.png" />-self-adjoint) operator corresponding to eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380027.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380028.png" />), does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380029.png" />.

The following theorem [[#References|[1]]] is fundamental in the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380031.png" />-self-adjoint operators on a Pontryagin space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380032.png" />: For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380033.png" />-self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380035.png" />) there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380036.png" />-dimensional (maximal) non-negative invariant subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380037.png" /> in which all eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380038.png" /> have non-negative imaginary parts, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380039.png" />-dimensional non-negative invariant subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380040.png" /> in which all eigen values have non-positive imaginary parts. A similar statement in which the upper (lower) half-plane is replaced by the exterior (interior) of the unit disc is also valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380041.png" />-unitary operators, and under certain additional conditions — even for operators on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380042.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380043.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380045.png" />-unitary operator, then its maximal invariant subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380047.png" /> can be chosen so that the elementary divisors of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380049.png" /> are of minimal order. In order that a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380050.png" /> with no roots inside the unit disc has the property: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380052.png" />, it is necessary and sufficient that it can be divided by the minimal annihilating polynomial of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380053.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380054.png" /> is a cyclic operator, then its non-negative invariant subspaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380055.png" /> are uniquely determined. In this case the above-mentioned property of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380056.png" /> with roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380057.png" /> outside the unit disc, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380058.png" />, is equivalent to the divisibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380059.png" /> by the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380060.png" />.

Each completely-continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380061.png" />-self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380062.png" /> on a Pontryagin space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380063.png" /> such that zero belongs to its continuous spectrum does not have a residual spectrum. The root vectors of such an operator form a [[Riesz basis|Riesz basis]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380064.png" /> with respect to the (definite) norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380065.png" />.

Many facts concerning invariant subspaces and the spectrum can be generalized to a case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380067.png" />-isometric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380069.png" />-non-expanding operators. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380070.png" /> is an arbitrary set of eigen values of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380071.png" />-isometric operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380073.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380074.png" /> is the order of the elementary divisor at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380075.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380076.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380077.png" />-non-expanding boundedly-invertible operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380078.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380079.png" />-dimensional invariant non-negative subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380080.png" /> such that all eigen values of the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380081.png" /> lie in the unit disc [[#References|[2]]]. A similar fact holds for maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380083.png" />-dissipative operators. In general, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380084.png" />-dissipative operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380086.png" />, has at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380087.png" /> eigen values in the upper half-plane. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380088.png" />-isometric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380089.png" />-symmetric (and more generally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380090.png" />-non-expanding and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380091.png" />-dissipative) operators are related by the Cayley transformation (cf. [[Cayley transform|Cayley transform]]), which has on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380092.png" /> all natural properties [[#References|[2]]]. This fact allows one to develop the extension theory simultaneously for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380093.png" />-isometric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380094.png" />-symmetric operators. In particular, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380095.png" />-isometric (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380096.png" />-symmetric) operator can be extended to a maximal one. If its deficiency indices are different, then it has no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380097.png" />-unitary (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380098.png" />-self-adjoint) extensions. If these indices are equal and finite, then any maximal extension is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p07380099.png" />-unitary (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800100.png" />-self-adjoint).

For completely-continuous operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800101.png" />, a number of statements on the completeness of the system of root vectors, analogous to the corresponding facts from the theory of dissipative operators on spaces with a definite metric, is valid.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   "Hermitian operators in a space with indefinite metric"  ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''8'''  (1944)  pp. 243–280  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.S. Iokhvidov,   M.G. Krein,   "Spectral theory of operators in a space with indefinite metric I"  ''Transl. Amer. Math. Soc. (2)'' , '''13'''  (1960)  pp. 105–175  ''Trudy Moskov. Mat. Obshch.'' , '''5'''  (1956)  pp. 367–432</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.S. Iokhvidov,   M.G. Krein,   "Spectral theory of operators in a space with indefinite metric II"  ''Trudy Moskov. Mat. Obshch.'' , '''8'''  (1959)  pp. 413–496  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T.Ya. Azizov,   I.S. Iokhvidov,   "Linear operators in Hilbert spaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800102.png" />-metric"  ''Russian Math. Surveys'' , '''26''' :  4  (1971)  pp. 45–97  ''Uspekhi Mat. Nauk'' , '''26''' :  4  (1971)  pp. 43–92</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.G. Krein,   "Introduction to the geometry of indefinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800103.png" />-spaces and the theory of operators in these spaces" , ''Second Math. Summer School'' , '''1''' , Kiev  (1965)  pp. 15–92  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.A. Naimark,   R.S. Ismagilov,   "Representations of groups and algebras in a space with indefinite metric"  ''Itogi Nauk. i Tekhn. Mat. Anal.''  (1969)  pp. 73–105  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Nagy,   "State vector spaces with indefinite metric in quantum field theory" , Noordhoff  (1966)</TD></TR></table>



====Comments====
Pontryagin spaces form a subclass of the class of Krein spaces (cf. [[Krein space|Krein space]] and also [[Hilbert space with an indefinite metric|Hilbert space with an indefinite metric]]). The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800104.png" /> appearing in the beginning of the main article above is the fundamental symmetry (see [[Krein space|Krein space]]), which defines the indefinite inner product via the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073800/p073800105.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.Ya. Azizov,   I.S. [I.S. Iokhvidov] Iohidov,   "Linear operators in spaces with an indefinite metric" , Wiley  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.S. [I.S. Iokhvidov] Iohidov,   M.G. Krein,   H. Langer,   "Introduction to the spectral theory of operators in spaces with an indefinite metric" , Akademie Verlag  (1982)</TD></TR></table>