Spectral theory
of linear operators
The branch of functional analysis in which one studies the structure of a linear operator on the basis of its spectral properties (such as the location of the spectrum, the behaviour of the resolvent and the asymptotics of its eigenvalues). By a description of the structure of an operator one usually means one of the following: the determination of an equivalent operator on a prescribed class of concrete (often functional) models; a specific method of reconstructing it from a class of simpler operators (for example, in the form of a direct sum or direct integral); the discovery of a basis in which the matrix of the operator has its simplest form, the proof of completeness of the system of root vectors; a complete description of the lattice of invariant subspaces; the identification of maximal chains of invariant subspaces (triangular representation); or the construction of a sufficiently wide functional calculus.
A very popular (and productive) idea in spectral theory is that of decomposing an operator into a direct sum of operators corresponding to a partition of its spectrum. The first results of this sort (for infinite-dimensional spaces) were obtained by F. Riesz (1909), who proposed the following construction. Let be a bounded linear operator on a Banach space
with spectrum
and resolvent
(that is,
,
). Then the formula
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where is an arbitrary contour enclosing
, defines a functional calculus on the algebra of germs of functions holomorphic in a neighbourhood of
. If
is an open-and-closed subset of
and
is the function equal to 1 on
and to
on
, then one obtains a projection operator
which commutes with
and satisfies
.
A more general spectral theory is based on the concept of a spectral subspace. The spectral manifold of corresponding to a closed subset
is defined as the set
of all vectors
that have a local resolvent in
(that is, an analytic
-valued function
satisfying the condition
,
); a spectral subspace is the closure of a spectral manifold. If any two local resolvents of the same vector coincide on the intersection of their domains of definition (this means that the local resolvent of the zero vector is equal to zero, which holds, for example, for all operators without eigenvalues), then one says that the operator has the unique extension property. In this case, for every
there is a local resolvent with maximal domain of definition, whose complement is called the local spectrum of
at the vector
and is written
. Thus, for an operator
possessing the unique extension property,
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if is closed, then
. In the general case the analogous statement about inclusion of spectral subspaces is false. Spectral subspaces satisfy the duality condition
(where
and
are disjoint closed sets), but the other natural condition
(where
are open and
) may be violated. This inclusion becomes valid if one replaces its right-hand side by the "weak spectral subspace"
(where
consists of the vectors
such that for every
there is an analytic
-valued function
with the property that
,
). Sufficient conditions for stronger separability of the spectrum are known. In particular, for operators with a real spectrum the restriction
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on the growth of the resolvent implies the existence (for any open covering of the spectrum) of a family of -invariant subspaces linearly generating
and such that the spectra of the restrictions of
to them are inscribed in the covering. In fact, such operators belong to the class of decomposable operators, defined as operators for which the spectral manifolds are closed and which satisfy the following condition: For any open covering
of the spectrum of
, the subspaces
linearly generate
. This class of operators contains all operators with resolvents that satisfy the condition of analytic majorizability (examples are compact operators, weak perturbations of spectral operators, multipliers of Fourier series in
, and
-symmetric operators), and it is stable under analytic mappings and (given certain restrictions) taking limits, and under the formation of restrictions and quotients. At the same time, an abundance of spectral subspaces (when the spectrum is sufficiently rich) guarantees the value of spectral theory. An example has been constructed of an operator which is beyond the limits of any given spectral decomposition, as the spectra of all its restrictions onto invariant subspaces coincide with the interval
.
Even in the case of a rarefied spectrum, the restrictions of an operator onto the spectral subspaces can have a fairly complicated structure (a fine structure). Thus, every pole of the resolvent is an eigenvalue whose ascent (the maximal length of a root chain) is equal to the order of the pole; the corresponding spectral subspace is a root subspace. In the case of operators on finite-dimensional spaces this reduces to the decomposition of the operator into a direct sum of Jordan cells constructed from root chains. Analogues of the Jordan form also occupy an important place in general spectral theory; the role of Jordan cells can be played by operators with a one-point spectrum and a cyclic vector, by operators with a linearly ordered lattice of invariant subspaces (such operators are called unicellular operators; among the operators on finite-dimensional spaces this property is possessed by Jordan cells only), or by operators having simple concrete representations (models). However, the existence of such a decomposition is not universal: there are operators whose lattice of invariant subspaces and spectrum are arranged in a way too complex for one to be able to regard them as elementary "cells" , and which at the same time do not possess even one pair of disjoint invariant subspaces. For a long time it was not known whether every bounded operator (on a space of a dimension greater than 1) possesses a non-trivial invariant subspace. A positive answer to this question has been obtained for compact operators, operators that commute with compact operators, operators that are close to Hermitian or unitary operators, subnormal operators, and operators belonging to certain other special classes. In 1984 C.J. Read [8] constructed examples of operators on certain Banach spaces (including ) without invariant subspaces. For reflexive spaces the problem is still open (1990).
Some results of finite-dimensional spectral theory have simple analogues in the spectral theory of compact operators. Thus, the spectrum of a compact operator is at most countable and its only possible accumulation point is , the non-zero points of the spectrum are poles of the resolvent, the root subspaces are finite-dimensional, and the adjoint operator has the same structure of restrictions onto the root subspaces. However, even in the case when the point spectrum is sufficiently rich and the root vectors of
span the whole of
(in such cases one says that
is a complete operator), the decomposition of
into a direct of sum of root subspaces may not be valid, due to geometric singularities of their mutual location.
If is a Hilbert space (in this case one writes
instead of
), then every compact operator
can be represented as the sum of a series
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that is,
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where is a non-increasing sequence of positive numbers and
,
are orthonormal systems. The numbers
are called the singular numbers, or
-numbers, of
; they coincide with the eigenvalues of the operator
, enumerated in decreasing order, taking multiplicities into account. Moreover,
, where
ranges over the set of projectors of co-rank
(a minimax characterization of singular numbers), and
coincides with the distance from
to the set of operators of rank
, which expresses numerically the correspondence between the rate of decrease of the singular numbers of the operator and its proximity to operators of finite rank. Based on this there are estimates for the singular numbers of sums and products, from which it follows that specific conditions on the rate of decrease of
-numbers distinguish ideals in the algebra of operators. In particular,
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is an ideal which, when , is a Banach space with respect to the norm
. The space
is a Hilbert space, and its elements are called Hilbert–Schmidt operators; for any
-realization of
there is a representation of all Hilbert–Schmidt operators as integral operators with square-summable kernels. Operators in
are called nuclear or trace-class operators: the trace defined on the ideal of operators of finite rank extends to a continuous functional on
whose value on any operator coincides with the sum of (the series of) the diagonal elements of its matrix, and also with the sum of its eigenvalues. For operators of the form
, where
, one can define the concept of a determinant (the infinite product of the eigenvalues). The function
is called the characteristic determinant of
. This is a natural generalization of the characteristic polynomial of a matrix, and since there are suitable estimates, it plays a useful role in the spectral theory of nuclear operators. In particular, the resolvent of an operator
is related to the characteristic determinant by the formula (E. Fredholm, 1903)
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where is an entire operator-function whose coefficients are expressed in terms of "partial traces" of
. The formulas and estimates for the resolvent obtained in this way carry over to operators in
,
(which is important in applications), and lead to the following tests of completeness: 1) if
, where
,
is compact and
, then
is complete (Keldysh' theorem; it has many applications in the spectral theory of differential operators); 2) if
and the range of values of the quadratic form
is contained in some angle of size
, then
is complete.
Compact operators whose spectra consist of the single point (a condition opposite to that of completeness) are called Volterra operators, in view of the fact that the Volterra integral operators
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are their prototypes. More precisely, every Hilbert–Schmidt Volterra operator is unitarily equivalent to a Volterra integral operator on a space of vector-functions; operators not belonging to have models whose kernels are generalized functions. Such integral representations are analogues of triangular representations of matrices. Techniques for integrating operator-functions by a chain of projectors have been developed and on this basis an abstract triangular representation has been obtained for a Volterra operator:
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where is a maximal chain of
-invariant projectors. This has led to a refinement and generalization of the basic theorem in the theory of integral representations, to a proof of important relations between the distributions of the eigenvalues of the Hermitian components of Volterra operators that are close to the identity, to the construction of triangular factorizations of operators, and to the establishment of a connection between spectral theory and certain questions in the theory of boundary value problems for canonical systems of differential equations (in particular, it has enabled one to use operator methods to investigate the question of stability of such systems).
The longstanding problem of the existence of chains of rank 1 for an arbitrary compact operator, i.e. of having a cyclic vector, has been solved in the negative. The existence of invariant chains of rank 1 has been proved for dissipative operators with a nuclear imaginary component, and as a result their triangular representations have a more complete form. There is also a theory of Jordan representations for such operators, and this is consistent with the classical (finite-dimensional) case: Every operator decomposes into a quasi-direct sum of unicellular operators, where the condition of being unicellular in this class of operators is equivalent to the existence of a cyclic vector. In this theory, a central role is played by the concept of a characteristic operator-function.
In close analogy with geometric constructions in the theory of unitary dilations, the concept of the characteristic operator-function of contraction (that is, of an operator whose norm does not exceed one) lies at the heart of the spectral theory of this class of operators. This characteristic operator-function of is a function
defined in the open unit disc
, taking values in the space of operators from
to
(where
) and satisfying the relation
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The characteristic operator-function is analytic in and is contractive:
. If
and
tend to zero in the strong operator topology (such operators form the class
), then
is an inner function, that is, its boundary values on
are almost-everywhere equal to 1. Conversely, for any inner operator-valued function
one can construct the contraction
for which
by restricting the operator of multiplication by
on the Hardy space
to the orthogonal complement
of the subspace
. This construction, called the functional contraction model, enables one to translate problems of spectral theory into the language of classical function theory, where they take the form of problems of interpolation, rational approximation, analytic continuation, and special factorization, among others. The functional model can be used to develop a richer functional calculus by defining for
the operator
as the restriction to
of the operator of multiplication by
(the condition that
is not necessary, for
it is important to be completely non-unitary). If this calculus is not injective, that is, if
for some function
,
, then
is called a contraction of class
. A contraction
possesses a minimal inner function
(the generator of the ideal of all functions annihilating
);
is an analogue of the minimal polynomial of a matrix: it determines many of the spectral properties of
. Thus, a contraction
is complete if and only if
is a Blaschke product (and in this case
admits spectral synthesis). The point spectrum
of a contraction
coincides with the set of zeros of
, and
is obtained from
by the adjunction of those points of the boundary
to which
cannot be analytically continued. The fact that the contractions in
have an at most countable spectrum in
indicates the restrictions of this class. On the other hand, it contains, for example, all contractions whose deficiency operators
are nuclear. If
are operators of rank 1, then the functional model operates on the classical Hardy space
and is completely determined by the scalar inner function
; in this case one writes
. The spectral theory of the contractions
is most closely akin to that of analytic functions and has been studied the most. These contractions play the role of Jordan cells in the spectral theory of contractions in
, in view of the fact that every contraction
is quasi-similar to a direct sum
. The more usual Jordan decomposition (into unicellular operators) for a
is not always possible.
References
[1a] | N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , 3 , Interscience (1971) MR0412888 |
[1b] | N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963) MR0188745 |
[2] | H. Radjavi, P. Rosenthal, "Invariant subspaces" , Springer (1973) MR0367682 Zbl 0269.47003 |
[3] | I. Colojoară, C. Foiaş, "Theory of generalized spectral operators" , Gordon & Breach (1968) MR0394282 Zbl 0189.44201 |
[4] | 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 |
[5] | 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 |
[6] | B. Szökefalvi-Nagy, Ch. Foiaş, "Harmonic analysis of operators on Hilbert space" , North-Holland (1970) (Translated from French) MR275190 |
[7] | N.K. Nikol'skii, "Treatise on the shift operator: spectral function theory" , Springer (1986) (Translated from Russian) Zbl 0587.47036 |
[8] | C.J. Read, "A solution to the invariant subspace problem" Bull. London Math. Soc. , 16 : 4 (1984) pp. 337–401 MR0749447 Zbl 0566.47003 |
Comments
For the notions of a root vector and root subspace of a linear operator cf. Root vector.
A root chain (corresponding to the root ) of a linear operator
is a sequence of non-zero vectors
such that
,
. It is also called a Jordan chain, [a1].
Quite generally, the ascent of a linear operator on a Banach space is defined as the smallest integer
such that
, and then
for all
. If no such integer
exists, the ascent
of
is set equal to
.
Let denote the range of an operator
on a Banach space
. The descent of a linear operator
is defined as the smallest integer such that
, and then
for all
. If no such
exists, the descent
of
is set equal to
.
If is a bounded linear operator and
and
are both finite, then
(
) and
. Especially for finite-dimensional Banach spaces this is known as the Fitting decomposition of
corresponding to
. The operator
correspondingly becomes a direct sum of a bijective operator
and a nilpotent operator
; this is called the Fitting decomposition of the operator
. Fitting's lemma applies also in other contexts. For instance, for a module endomorphism
of a module
of finite length there is an
such that
.
References
[a1] | M.S. Birman, M.Z. Solomyak, "Spectral theory of selfadjoint operators in Hilbert space" , Reidel (1987) pp. Chapt. 3, §5 (Translated from Russian) MR1192782 |
[a2] | H.R. Dowson, "Spectral theory of linear operators" , Acad. Press (1978) MR0511427 Zbl 0384.47001 |
Spectral theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_theory&oldid=24566