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 $ T $ be a bounded linear operator on a Banach space $ X $ with spectrum $ \sigma ( T) $ and resolvent $ R _ {T} ( \lambda ) $( that is, $ R _ {T} ( \lambda ) = ( T - \lambda I ) ^ {-1} $, $ \lambda \in \mathbf C \setminus \sigma ( T) $). Then the formula

$$ f ( T) = ( 2 \pi i ) ^ {-1} \oint _ \Gamma f ( \lambda ) R _ {T} ( \lambda ) d \lambda , $$

where $ \Gamma $ is an arbitrary contour enclosing $ \sigma ( T) $, defines a functional calculus on the algebra of germs of functions holomorphic in a neighbourhood of $ \sigma ( T) $. If $ \delta $ is an open-and-closed subset of $ \sigma ( T) $ and $ f $ is the function equal to 1 on $ \delta $ and to $ 0 $ on $ \sigma ( T) \setminus \delta $, then one obtains a projection operator $ P _ {T} ( \delta ) $ which commutes with $ T $ and satisfies $ \sigma ( T \mid _ {P _ {T} ( \delta ) X } ) = \delta $.

A more general spectral theory is based on the concept of a spectral subspace. The spectral manifold of $ T $ corresponding to a closed subset $ \delta \subset \sigma ( T) $ is defined as the set $ X _ {T} ( \delta ) $ of all vectors $ x \in X $ that have a local resolvent in $ \mathbf C \setminus \delta $( that is, an analytic $ X $- valued function $ f ( \lambda ) $ satisfying the condition $ ( T - \lambda I ) f ( \lambda ) = x $, $ \lambda \in \mathbf C \setminus \delta $); 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 $ x \in X $ there is a local resolvent with maximal domain of definition, whose complement is called the local spectrum of $ T $ at the vector $ x $ and is written $ \sigma ( T , x ) $. Thus, for an operator $ T $ possessing the unique extension property,

$$ X _ {T} ( \delta ) = \\{ {x \in X } : { \sigma ( T , x ) \subset \delta } \\} ; $$

if $ X _ {T} ( \delta ) $ is closed, then $ \sigma ( T \mid _ {X _ {T} ( \delta ) } ) \subset \delta $. In the general case the analogous statement about inclusion of spectral subspaces is false. Spectral subspaces satisfy the duality condition $ X _ {T} ( \delta _ {1} ) ^ \perp \supset X _ {T*} ( \delta _ {2} ) $( where $ \delta _ {1} $ and $ \delta _ {2} $ are disjoint closed sets), but the other natural condition $ X _ {T} ( \overline{G}\; ) ^ \perp \subset \overline{ {X _ {T*} ( \overline{G}\; _ {2} ) }}\; $( where $ G _ {1} , G _ {2} $ are open and $ G _ {1} \cup G _ {2} = \sigma ( T) $) may be violated. This inclusion becomes valid if one replaces its right-hand side by the "weak spectral subspace" $ X _ {T} ^ {W} ( \overline{G}\; _ {2} ) $( where $ X _ {T} ^ {W} ( \delta ) $ consists of the vectors $ x \in X $ such that for every $ \epsilon > 0 $ there is an analytic $ X $- valued function $ f _ \epsilon ( \lambda ) $ with the property that $ \| ( T - \lambda I ) f _ \epsilon ( \lambda ) - x \| \leq \epsilon $, $ \lambda \in \mathbf C \setminus \delta $). Sufficient conditions for stronger separability of the spectrum are known. In particular, for operators with a real spectrum the restriction

$$ \int\limits _ { 0 } \mathop{\rm log} ^ {+} \mathop{\rm log} ^ {+} \left ( \sup _ { s } \| R _ {T} ( s + i t ) \| \right ) d t < \infty $$

on the growth of the resolvent implies the existence (for any open covering of the spectrum) of a family of $ T $- invariant subspaces linearly generating $ X $ and such that the spectra of the restrictions of $ T $ 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 $ \\{ G _ {i} \\}_{i=1}^ {n} $ of the spectrum of $ T $, the subspaces $ X _ {T} ( \overline{G}\; _ {i} ) $ linearly generate $ X $. 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 $ l _ {p} $, and $ J $- 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 $ [ 0 , 1 ] $.

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 $ l _ {1} $) 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 $ 0 $, 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 $ T $ span the whole of $ X $( in such cases one says that $ T $ is a complete operator), the decomposition of $ X $ into a direct of sum of root subspaces may not be valid, due to geometric singularities of their mutual location.

If $ X $ is a Hilbert space (in this case one writes $ H $ instead of $ X $), then every compact operator $ T \in {\mathcal L} ( H) $ can be represented as the sum of a series

$$ \sum _ { n } s _ {n} f _ {n} \otimes e _ {n} , $$

that is,

$$ T x = \sum _ { n } s _ {n} ( x , f _ {n} ) e _ {n} ,\ \ x \in H , $$

where $ \\{ s _ {n} \\} $ is a non-increasing sequence of positive numbers and $ \\{ f _ {n} \\} $, $ \\{ e _ {n} \\} $ are orthonormal systems. The numbers $ s _ {n} = s _ {n} ( T) $ are called the singular numbers, or $ s $- numbers, of $ T $; they coincide with the eigenvalues of the operator $ ( T T ^ {*} ) ^ {1/2} $, enumerated in decreasing order, taking multiplicities into account. Moreover, $ s _ {n} ( T) = \inf \| T P \| $, where $ P $ ranges over the set of projectors of co-rank $ n $( a minimax characterization of singular numbers), and $ s _ {n} ( T) $ coincides with the distance from $ T $ to the set of operators of rank $ n $, 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 $ s $- numbers distinguish ideals in the algebra of operators. In particular,

$$ \gamma _ {p} = \\{ {T } : {| T | _ {p} \equiv ( \sum s _ {n} ^ {p} ( T) ) ^ {1/p} < \infty } \\} $$

is an ideal which, when $ p \geq 1 $, is a Banach space with respect to the norm $ | T | _ {p} $. The space $ \gamma _ {2} $ is a Hilbert space, and its elements are called Hilbert–Schmidt operators; for any $ L _ {2} $- realization of $ H $ there is a representation of all Hilbert–Schmidt operators as integral operators with square-summable kernels. Operators in $ \gamma _ {1} $ are called nuclear or trace-class operators: the trace defined on the ideal of operators of finite rank extends to a continuous functional on $ \gamma _ {1} $ 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 $ I + T $, where $ T \in \gamma _ {1} $, one can define the concept of a determinant (the infinite product of the eigenvalues). The function $ \mathop{\rm det} ( I - \mu T ) $ is called the characteristic determinant of $ T $. 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 $ T \in \gamma _ {1} $ is related to the characteristic determinant by the formula (E. Fredholm, 1903)

$$ R _ {T} ( \lambda ) = F _ {T} ( \lambda ^ {-1} ) \mathop{\rm det} \ ( I - \lambda ^ {-1} T ) ^ {-1} , $$

where $ F _ {T} $ is an entire operator-function whose coefficients are expressed in terms of "partial traces" of $ T $. The formulas and estimates for the resolvent obtained in this way carry over to operators in $ \gamma _ {p} $, $ p > 1 $( which is important in applications), and lead to the following tests of completeness: 1) if $ T = A ( I + S ) $, where $ A = A ^ {*} \in \gamma _ {p} $, $ S $ is compact and $ \mathop{\rm Ker} A = 0 $, then $ T $ is complete (Keldysh' theorem; it has many applications in the spectral theory of differential operators); 2) if $ T \in \gamma _ {p} $ and the range of values of the quadratic form $ ( T x , x ) $ is contained in some angle of size $ \pi / p $, then $ T $ is complete.

Compact operators whose spectra consist of the single point $ \lambda = 0 $( a condition opposite to that of completeness) are called Volterra operators, in view of the fact that the Volterra integral operators

$$ T f ( x) = \int\limits _ { 0 } ^ { x } K ( x , y ) f ( y) d y $$

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 $ \gamma _ {2} $ 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:

$$ T = \int\limits _ {\mathcal P} P ( T - T ^ {*} ) d P , $$

where $ {\mathcal P} $ is a maximal chain of $ T $- 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 $ T $ is a function $ \theta _ {T} ( \lambda ) $ defined in the open unit disc $ \Delta \in \mathbf C $, taking values in the space of operators from $ \overline{ {D _ {T} ( H) }}\; $ to $ \overline{ {D _ {T*} ( H) }}\; $( where $ D _ {T} = ( I - T ^ {*} T ) ^ {1/2} $) and satisfying the relation

$$ \theta _ {T} ( \lambda ) D _ {T} = \ D _ {T*} ( I - \lambda T ^ {*} ) ^ {-1} ( I \lambda - T ) . $$

The characteristic operator-function is analytic in $ \Delta $ and is contractive: $ \| \theta _ {T} ( \lambda ) \| \leq 1 $. If $ T ^ {n} $ and $ T ^ {*} n $ tend to zero in the strong operator topology (such operators form the class $ C _ {00} $), then $ \theta _ {T} $ is an inner function, that is, its boundary values on $ \partial \Delta $ are almost-everywhere equal to 1. Conversely, for any inner operator-valued function $ \theta : \Delta \rightarrow {\mathcal B} ( E _ {1} , E _ {2} ) $ one can construct the contraction $ T $ for which $ \theta _ {T} = \theta $ by restricting the operator of multiplication by $ \lambda $ on the Hardy space $ H _ {E _ {2} } ^ {2} ( \Delta ) $ to the orthogonal complement $ K _ \theta $ of the subspace $ \theta H _ {E _ {1} } ^ {2} $. 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 $ \phi \in H ^ \infty ( \Delta ) $ the operator $ \phi ( T) $ as the restriction to $ K _ \theta $ of the operator of multiplication by $ \phi ( \lambda ) $( the condition that $ T \in C _ {00} $ is not necessary, for $ T $ it is important to be completely non-unitary). If this calculus is not injective, that is, if $ \phi ( T) = 0 $ for some function $ \phi \in H ^ \infty $, $ \phi \neq 0 $, then $ T $ is called a contraction of class $ C _ {0} $. A contraction $ T \in C _ {0} $ possesses a minimal inner function $ m _ {T} $( the generator of the ideal of all functions annihilating $ T $); $ m _ {T} $ is an analogue of the minimal polynomial of a matrix: it determines many of the spectral properties of $ T $. Thus, a contraction $ T \in C _ {0} $ is complete if and only if $ m _ {T} $ is a Blaschke product (and in this case $ T $ admits spectral synthesis). The point spectrum $ \sigma _ {p} ( T) $ of a contraction $ T \in C _ {0} $ coincides with the set of zeros of $ m _ {T} $, and $ \sigma ( T) $ is obtained from $ \sigma _ {p} ( T) $ by the adjunction of those points of the boundary $ \partial \Delta $ to which $ m _ {T} $ cannot be analytically continued. The fact that the contractions in $ C _ {0} $ have an at most countable spectrum in $ \Delta $ indicates the restrictions of this class. On the other hand, it contains, for example, all contractions whose deficiency operators $ D _ {T} , D _ {T*} $ are nuclear. If $ D _ {T} , D _ {T*} $ are operators of rank 1, then the functional model operates on the classical Hardy space $ H ^ {2} ( \Delta ) $ and is completely determined by the scalar inner function $ m = m _ {T} = \theta _ {T} $; in this case one writes $ T = S ( m) $. The spectral theory of the contractions $ S ( m) $ 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 $ C _ {0} $, in view of the fact that every contraction $ T \in C _ {0} $ is quasi-similar to a direct sum $ \oplus_{i=1}^ {N} S ( m _ {1} ) $. The more usual Jordan decomposition (into unicellular operators) for a $ T \in C _ {0} $ is not always possible.

#### References[edit]

[1a] |  N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , 3 , Interscience (1971) MR0412888

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[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[edit]

For the notions of a root vector and root subspace of a linear operator cf. Root vector.

A root chain (corresponding to the root $ \xi $) of a linear operator $ A $ is a sequence of non-zero vectors $ x _ {0} \dots x _ {n} $ such that $ Ax _ {0} = \xi x _ {0} $, $ A x _ {1} = \xi x _ {1} + x _ {0} \dots A x _ {n} = \xi x _ {n} + x _ {n-1} $. It is also called a Jordan chain, [a1].

Quite generally, the ascent of a linear operator $ A $ on a Banach space is defined as the smallest integer $ n $ such that $ \mathop{\rm Ker} ( A ^ {n} ) = \mathop{\rm Ker} ( A ^ {n+1} ) $, and then $ \mathop{\rm Ker} ( A ^ {n} ) = \mathop{\rm Ker} ( A ^ {n+k} ) $ for all $ k \geq 0 $. If no such integer $ n $ exists, the ascent $ \alpha ( A) $ of $ A $ is set equal to $ \infty $.

Let $ R( A) = AX $ denote the range of an operator $ A $ on a Banach space $ X $. The descent of a linear operator $ A $ is defined as the smallest integer such that $ R( A ^ {n} ) = R( A ^ {n+1} ) $, and then $ R( A ^ {n} ) = R( A ^ {n+k} ) $ for all $ k \geq 0 $. If no such $ n $ exists, the descent $ \delta ( A) $ of $ A $ is set equal to $ \infty $.

If $ A $ is a bounded linear operator and $ \alpha ( A) $ and $ \delta ( A) $ are both finite, then $ \alpha ( A) = \delta ( A) $( $ = p $) and $ X = R( A ^ {p} ) \oplus \mathop{\rm Ker} ( A ^ {p} ) $. Especially for finite-dimensional Banach spaces this is known as the Fitting decomposition of $ X $ corresponding to $ A $. The operator $ A $ correspondingly becomes a direct sum of a bijective operator $ A _ {1} : R( A ^ {p} )\rightarrow R( A ^ {p} ) $ and a nilpotent operator $ A _ {2} : \mathop{\rm Ker} ( A ^ {p} ) \rightarrow \mathop{\rm Ker} ( A ^ {p} ) $; this is called the Fitting decomposition of the operator $ A $. Fitting's lemma applies also in other contexts. For instance, for a module endomorphism $ \alpha $ of a module $ M $ of finite length there is an $ n $ such that $ M = \mathop{\rm Im} ( \alpha ^ {n} ) \oplus \mathop{\rm Ker} ( \alpha ^ {n} ) $.

#### References[edit]

[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

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[a2] |  H.R. Dowson, "Spectral theory of linear operators" , Acad. Press (1978) MR0511427 Zbl 0384.47001