Eigenvalues of non-selfadjoint operators: A comparison of two approaches
The central problem we consider is the distribution of eigenvalues of closed linear operators which are not selfadjoint, with a focus on those operators which are obtained as perturbations of selfadjoint linear operators. Two methods are explained and elaborated. One approach uses complex analysis to study a holomorphic function whose zeros can be identified with the eigenvalues of the linear operator. The second method is an operator theoretic approach involving the numerical range. General results obtained by the two methods are derived and compared. Applications to non-selfadjoint Jacobi and Schrödinger operators are considered. Some possible directions for future research are discussed.
The importance of eigenvalues and eigenvectors is clear to every student of mathematics, science or engineering. As a simple example, consider a linear dynamical system which is described by an equation of the form
where is an element in a linear space and a linear operator in . If we can find an eigenpair , with , then we have solved (1.1) with the initial condition : . If we can find a whole basis of eigenvectors, we have solved (1.1) for any initial condition by decomposing with respect to this basis. So the knowledge of the eigenvalues of (or more generally, the analysis of its spectrum) is essential for the understanding of the corresponding system.
The spectral analysis of linear operators has a quite long history, as everybody interested in the field is probably aware of. Still, we think that it can be worthwhile to begin this introduction with a short historical survey, which will also help to put the present article in its proper perspective. The origins of spectral analysis can be traced back at least as far as the work of D’Alembert and Euler (1740-50’s) on vibrating strings, where eigenvalues correspond to frequencies of vibration, and eigenvectors correspond to modes of vibration. When the vibrating string’s density and tension is not uniform, the eigenvalue problem involved becomes much more challenging, and an early landmark of spectral theory is Sturm and Liouville’s (1836-1837) analysis of general one-dimensional problems on bounded intervals, showing the existence of an infinite sequence of eigenvalues. This naturally gave rise to questions about corresponding results for differential operators on higher-dimensional domains, with the typical problem being the eigenvalues of the Laplacian on a bounded domain with Dirichlet boundary conditions. The existence of the first eigenvalue for this problem was obtained by Schwartz (1885), and of the the second eigenvalue by Picard (1893), and it was Poincaré (1894) who obtained existence of all eigenvalues and their basic properties. Inspired by Poincarés work, Fredholm (1903) undertook the study of the spectral theory of integral operators. Hilbert (1904-1910), generalizing the work of Fredholm, introduced the ideas of quadratic forms on infinite dimensional linear spaces and of completely continuous forms (compact operators in current terminology). He also realized that spectral analysis cannot be performed in terms of eigenvalues alone, developing the notion of continuous spectrum, which was prefigured in Wirtinger’s (1897) work on Hill’s equation. Weyl’s (1908) work on integral equations on unbounded intervals further stresses the importance of the continuous spectrum. The advent of quantum mechanics, formulated axiomatically by von Neumann (1927), who was the first to introduce the notion of an abstract Hilbert space, brought selfadjoint operators into the forefront of interest. Kato’s  rigorous proof of the selfadjointness of physically relevant Schrödinger operators was a starting point for the mathematical study of particular operators. In the context of quantum mechanics, eigenvalues have special significance, as they correspond to discrete energy levels, and thus form the basis for the quantization phenomenon, which in the pre-Schrödinger quantum theory had to be postulated a-priori. In recent years, non-selfadjoint operators are also becoming increasingly important in the study of quantum mechanical systems, as they arise naturally in, e.g., the optical model of nuclear scattering or the study of the behavior of unstable lasers (see  and references therein).
As this brief sketch111The interested reader can find much more information (and detailed references) in Mawhin’s account  on the origins of spectral analysis. of some highlights of the (early) history of spectral theory shows, eigenvalues, eigenvectors, and the spectrum provide an endless source of fascination for both mathematicians and physicists. At the most general level one may ask, given a class of linear operators (which in our case will always operate in a Hilbert space), what can be said about the spectrum of operators ? Of course, the more restricted is the class of operators considered the more we can say, and the techniques available for studying different classes can vary enormously. For example, an important part of the work of Hilbert is a theory of selfadjoint compact operators, which in particular characterizes their spectrum as an infinite sequence of real eigenvalues. Motivated by various applications, this class of operators can be restricted or broadened to yield other classes worth studying. For example, the study of eigenvalues of the Dirichlet problem in a bounded domain is a restriction of the class of compact selfadjoint eigenvalue problems, which yields a rich theory relating the eigenvalues to the geometrical properties of the domain in question. As far as broadening the class of operators goes, one can consider selfadjoint operators which are not compact, leading to a vast domain of study which is of great importance to a variety of areas of application, perhaps the most prominent being quantum mechanics. One can also consider compact operators which are not selfadjoint (and which might act in general Banach spaces), leading to a field of research in which natural sub-classes of the class of compact operators are defined and their sets of eigenvalues are studied (see e.g. the classical works of Gohberg and Krein  or Pietsch ).
One may also lift both the assumption of selfadjointness and that of compactness. However, some restriction on the class of operators considered must be made in order to be able to say anything nontrivial about the spectrum. The classes of operators that we will be considering here are those that arise by perturbing bounded or unbounded (in most cases selfadjoint) operators with no isolated eigenvalues by operators which are (relatively) compact, for example operators of the form , where is a bounded operator with spectrum and is a compact operator in a certain Schatten class. More precisely, we will be interested in the isolated eigenvalues of such operators and in their rate of accumulation to the essential spectrum . We will study this rate by analyzing eigenvalue moments of the form
where is the set of discrete eigenvalues, and by bounding these moments in terms of the Schatten norm of the perturbation .
It is well known that the summation of two ‘simple’ operators can generate an operator whose spectrum is quite difficult to understand, even in case that both operators are selfadjoint. In our case, at least one of the operators will be non-selfadjoint, so the huge toolbox of the selfadjoint theory (containing, e.g., the spectral theorem, the decomposition of the spectrum into its various parts or the variational characterization of the eigenvalues) will not be available. This will make the problem even more demanding and also indicates that we cannot expect to obtain as much information on the spectrum as in the selfadjoint case. At this point we cannot resist quoting E. B. Davies, who in the preface of his book  on the spectral theory of non-selfadjoint operators described the differences between the selfadjoint and the non-selfadjoint theory: ”Studying non-selfadjoint operators is like being a vet rather than a doctor: one has to acquire a much wider range of knowledge, and to accept that one cannot expect to have as high a rate of success when confronted with particular cases”.
In our previous work, which we review in this paper, we have developed and explored two quite different approaches to obtain results on the distribution of eigenvalues of non-selfadjoint operators. One approach, which has also benefitted from (and relies heavily on) some related work of Borichev, Golinskii and Kupin , involves the construction of a holomorphic function whose zeros coincide with the eigenvalues of the operator of interest (the ‘perturbation determinant’) and the study of these zeros by employing results of complex analysis. The second is an operator-theoretic approach using the concept of numerical range. One of our main aims in this paper is to present these two methods side by side, and to examine the advantages of each of them in terms of the results they yield. We shall see that each of these methods has certain advantages over the other.
The plan of this paper is as follows. In Chapter 2 we recall fundamental concepts and results of functional analysis and operator theory that will be used. In Chapter 3 we discuss results on zeros of complex functions that will later be used to obtain results on eigenvalues. In particular, we begin this chapter with a short explanation why results from complex analysis can be used to obtain estimates on eigenvalue moments of the form (1.2) in the first place. Next, in Chapter 4, we develop the complex-analysis approach to obtaining results on eigenvalues of perturbed operators, obtaining results of varying degrees of generality for Schatten-class perturbations of selfadjoint bounded operators and for relatively-Schatten perturbations of non-negative operators. A second, independent, approach to obtaining eigenvalue estimates via operator-theoretic arguments is exposed in Chapter 5, and applied to the same classes of operators. In Chapter 6 we carry out a detailed comparison of the results obtained by the two approaches in the context of Schatten-perturbations of bounded selfadjoint operators. In Chapter 7 we turn to applications of the results obtained in Chapter 4 and 5 to some concrete classes of operators, which allows us to further compare the results obtained by the two approaches in these specific contexts. We obtain results on the eigenvalues of Jacobi operators and of Schrödinger operators with complex potentials. These case-studies also give us the opportunity to compare the results obtained by our methods to results which have been obtained by other researchers using different methods. These comparisons give rise to some conjectures and open questions which we believe could stimulate further research. Some further directions of ongoing work related to the work discussed in this paper, and issues that we believe are interesting to address, are discussed in Chapter 8.
In this chapter we will introduce and review some basic concepts of operator and spectral theory, restricting ourselves to those aspects of the theory which are relevant in the later parts of this work. We will also use this chapter to set our notation and terminology. As general references let us mention the monographs of Davies , Gohberg, Goldberg and Kaashoek , Gohberg and Krein  and Kato .
2.1. The spectrum of linear operators
Let denote a complex separable Hilbert space and let be a linear operator in . The domain, range and kernel of are denoted by , and , respectively . We say that is an operator on if . The algebra of all bounded operators on is denoted by . Similarly, denotes the class of all closed operators in .
In the following we assume that is a closed operator in . The resolvent set of is defined as
and for we define
The complement of in , denoted by , is called the spectrum of . Note that is an open and is a closed subset of . We say that is an eigenvalue of if is nontrivial.
The extended resolvent set of is defined as
In particular, if we regard as a subset of the extended complex plane . Setting if , the operator-valued function
called the resolvent of , is analytic on . Moreover, for every the resolvent satisfies the inequality , where denotes the norm of 333We will use the same symbol to denote the norm on . and we agree that . Actually, if is a normal operator (that is, an operator commuting with its adjoint) then the spectral theorem implies that
If is an isolated point of the spectrum, we define the Riesz projection of with respect to by
where the contour is a counterclockwise oriented circle centered at , with sufficiently small radius (excluding the rest of ). We recall that a subspace is called -invariant if . In this case, denotes the restriction of to and the range of is a subspace of .
Proposition 2.1.1 (see, e.g., , p.326).
Let and let be isolated. If is defined as above, then the following holds:
is a projection, i.e., .
and are -invariant.
and is bounded.
We say that is a discrete eigenvalue if is an isolated point of and is of finite rank (in the literature these eigenvalues are also referred to as ”eigenvalues of finite type”). Note that in this case is indeed an eigenvalue of since and is -invariant and finite-dimensional. The positive integer
is called the algebraic multiplicity of with respect to . It has to be distinguished from the geometric multiplicity, which is defined as the dimension of the eigenspace (and so can be smaller than the algebraic multiplicity).
In this article only algebraic multiplicities will be considered and we will use the term ”multiplicity” as a synonym for ”algebraic multiplicity”.
The discrete spectrum of is now defined as
We recall that a linear operator is a Fredholm operator if it has closed range and both its kernel and cokernel are finite-dimensional. Equivalently, if is densely defined, then is Fredholm if it has closed range and both and are finite-dimensional. The essential spectrum of is defined as
Note that and that is a closed set.
For later purposes we will need the following result about the spectrum of the resolvent of .
Suppose that with . If , then
The same identity holds when, on both sides, is replaced by and , respectively. More precisely, is an isolated point of if and only if is an isolated point of and in this case
In particular, the algebraic multiplicities of and coincide.
We note that if and only if . Moreover, if is densely defined, then
The following proposition shows that the essential and the discrete spectrum of a linear operator are disjoint.
If and is an isolated point of , then if and only if . In particular,
While the spectrum of a selfadjoint operator can always be decomposed as
where the symbol denotes a disjoint union, the same need not be true in the non-selfadjoint case. For instance, considering the shift operator acting on , we have and , while , see , p.237-238. The following result gives a suitable criterion for the discreteness of the spectrum in the complement of .
Proposition 2.1.6 (, p.373).
Let and let be open and connected. If , then .
Hence, if is a (maximal connected) component of , then either
(in particular, ), or
and consists of an at most countable sequence of discrete eigenvalues which can accumulate at only.
A direct consequence of Proposition 2.1.6 is
Let with and assume that there are points of in both the upper and lower half-planes. Then .
We conclude this section with some remarks on the numerical range of a linear operator and its relation to the spectrum, see ,  for extensive accounts on this topic. The numerical range of is defined as
It was shown by Hausdorff and Toeplitz (see, e.g.,  Theorem 9.3.1) that the numerical range is always a convex subset of . Furthermore, if the complement of the closure of the numerical range is connected and contains at least one point of the resolvent set of , then and
Clearly, if then . Moreover, if is normal then the closure of coincides with the convex hull of , i.e. the smallest convex set containing .
2.2. Schatten classes and determinants
An operator is called compact if it is the norm limit of finite rank operators. The class of all compact operators forms a two-sided ideal in , which we denote by . The non-zero elements of the spectrum of are discrete eigenvalues. In particular, the only possible accumulation point of the spectrum is , and itself may or may not belong to the spectrum. More precisely, if is infinite-dimensional, as will be the case in most of the applications below, then .
For every we can find (not necessarily complete) orthonormal sets and in , and a set of positive numbers with , such that
Here the numbers are called the singular values of . They are precisely the eigenvalues of , in non-increasing order.
The Schatten class of order (with ), denoted by , consists of all compact operators on whose singular values are -summable, i.e.
We remark that is a linear subspace of for every and for we can make it into a complete normed space by setting
Note that for this definition provides only a quasi-norm. For consistency we set .
For we have the (strict) inclusion and
Similar to the class of compact operators, is a two-sided ideal in the algebra and for and we have
Moreover, if then and .
The following estimate is a Schatten class analog of Hölder’s inequality (see , p.88): Let and where . Then , where , and
While the singular values of a selfadjoint operator are just the absolute values of its eigenvalues, in general the eigenvalues and singular values need not be related. However, we have the following result of Weyl.
Let where and let denote its sequence of nonzero eigenvalues (counted according to their multiplicity). Then
In the remaining part of this section we will introduce the notion of an infinite determinant. To this end, let , where , and let denote its sequence of nonzero eigenvalues, counted according to their multiplicity and enumerated according to decreasing absolute value. The -regularized determinant of , where denotes the identity operator on , is
Here the convergence of the products on the right-hand side follows from (2.2.6).
It is clear from the definition that is invertible if and only if . Moreover, . Since the nonzero eigenvalues of and coincide () we have
if both .
The regularized determinant is a continuous function of . If is open and depends holomorphically on , then is holomorphic on . For a proof of both results we refer to .
We can define the perturbation determinant for non-integer valued Schatten classes as well: Since where , the -regularized determinant of is well defined, and so the above results can still be applied. Moreover, this determinant can be estimated in terms of the th Schatten norm of (see , ,  ): If , where , then
where is some positive constant.
2.3. Perturbation theory
The aim of perturbation theory is to obtain information about the spectrum of some operator by showing that it is close, in a suitable sense, to an operator whose spectrum is already known. In this case one can hope that some of the spectral characteristics of are inherited by . For instance, the classical Weyl theorem (see Theorem 2.3.4 below) implies the validity of the following result (also sometimes called Weyl’s Theorem).
Let with . If the resolvent difference is compact for some , then .
If is compact for some , then the same is true for every . This is a consequence of the Hilbert-identity
valid for .
Let and let be selfadjoint. Suppose that there are points of in both the upper and lower half-planes. If for some , then and
In the following we will study perturbations of the form , understood as the usual operator sum defined on . More precisely, we assume that has non-empty resolvent set and that is a relatively bounded perturbation of , i.e. and there exist such that
for all . The infimum of all constants for which a corresponding exists such that the last inequality holds is called the -bound of . The operator is closed if the -bound of is smaller one. Note that is -bounded if and only if and for some , and the -bound is not larger than . The operator is called -compact if and for some . Every -compact operator is -bounded and the corresponding -bound is . Moreover, if is -compact and is Fredholm, then also is Fredholm (see, e.g., , p.238). The last implication is the main ingredient in the proof of Weyl’s theorem:
Let where and is -compact. Then .
2.4. Perturbation determinants
We have seen in the last section that the essential spectrum is stable under (relatively) compact perturbations. In this section, we will have a look at the discrete spectrum and construct a holomorphic function whose zeros coincide with the discrete eigenvalues of the corresponding operator. Throughout we make the following assumption.
and are closed densely defined operators in such that
for some and some fixed .
By Proposition 2.3.3, assumption (iii) follows from assumption (ii) if is selfadjoint with and if there exist points of in both the upper and lower half-planes. If and are bounded operators on then the second resolvent identity implies that assumption (ii) is equivalent to .
We begin with the case when : Then for we have
so if and only if is invertible. As we know from Section 2.2, this operator is invertible if and only if
By Assumption 2.4.1 we have , so we have shown that if and only if is a zero of the analytic function
For later purposes we note that .
so we can apply the previous discussion to the operators and , i.e. the function
is analytic on and
Note that, as above, we have .
We summarize the previous discussion in the following proposition.
We call the function the th perturbation determinant of by (the -dependence of is neglected in our notation). Without proof we note that the algebraic multiplicity of coincides with the order of as a zero of , see , p.20-22.
We conclude this section with some estimates.
Apply estimate (2.2.9). ∎
Let satisfy Assumption 2.4.1. Then for we have
If, in addition, where are bounded operators on such that for every , then for we have
While the non-zero eigenvalues of and coincide, the same need not be true for their singular values. In particular, while is automatically satisfied if satisfy Assumption 2.4.1, in general this need not imply that as well.
3. Zeros of holomorphic functions
In this chapter we discuss results on the distribution of zeros of holomorphic functions, which will subsequently be applied to the holomorphic functions defined by perturbation determinants to obtain results on the distribution of eigenvalues for certain classes of operators. We begin with a motivating discussion in Section 3.1, introducing the class of functions on the unit disk which will be our special focus of study. In Section 3.2 we consider results that can be obtained using the classical Jensen identity. In Section 3.3 we present the recent results of Borichev, Golinskii and Kupin and show that, for the class of functions that we are interested in, they yield more information than provided by the application of the Jensen identity.
3.1. Motivation: the complex analysis method for studying eigenvalues
We have seen in Section 2.4 that the discrete spectrum of a linear operator satisfying Assumption 2.4.1 coincides with the zero set of the corresponding perturbation determinant, which is a holomorphic function defined on the resolvent set of the ‘unperturbed’ operator . Moreover, we have a bound on the absolute value of this holomorphic function in the form of Propositions 2.4.5 and 2.4.6. Thus, general results providing information about the zeros of holomorphic functions satisfying certain bounds may be exploited to obtain information about the eigenvalues of the operator . This observation is the basis of the following complex-analysis approach to studying eigenvalues.
As an example, we consider the following situation: is assumed to be a selfadjoint operator with
where , and
where for some fixed . Given these assumptions, the spectrum of can differ from the spectrum of by an at most countable set of discrete eigenvalues, whose points of accumulation are contained in the interval . Moreover, is precisely the zero set of the th perturbation determinant defined by
It should therefore be possible to obtain further information on the distribution of the eigenvalues of by studying the analytic function , in particular, by taking advantage of the estimate provided on in Proposition 2.4.6, i.e.,
as well as the fact that . Note that the right-hand side of (3.1.2) is finite for any , but as approaches it can ‘explode’. A simple way to estimate the right-hand side of (3.1.2) from above and thus to obtain a more concrete estimate, is to use the identity
which is valid since is selfadjoint, and the inequality (2.2.5) to obtain
The inequality (3.1.4) is the best that we can obtain at a general level, that is without imposing any further restrictions on the operators and . However, as we shall show in Chapter 7.1, for concrete operators it is possible to obtain better inequalities by a more precise analysis of the -norm of . These inequalities will take the general form
where and are some non-negative parameters with . Note that (3.1.5) can be stronger than (3.1.4) in the sense that the growth of as approaches a point is estimated from above by , which can be smaller than the bound given by (3.1.4) (since if ). A similar remark applies to approaching one of the endpoints (since, e.g., if ). As we shall see, such differences are very significant in terms of the estimates on eigenvalues that are obtained.
The question then becomes how to use inequalities of the type (3.1.4), (3.1.5) to deduce information about the zeros of the holomorphic function . The study of zeros of holomorphic functions is, of course, a major theme in complex analysis. Since the holomorphic functions which we will be looking at will be defined on domains that are conformally equivalent to the open unit disk , we are specifically interested in results about zeros of functions , the class of holomorphic functions in the unit disk. Indeed, if is a domain which is conformally equivalent to the unit disk, we choose a conformal map so that the study of the zeros of the holomorphic function is converted to the study of the zeros of the function , where, denoting by the set of zeros of a holomorphic function , we have
We can also choose the conformal mapping so that , which implies that .
This conversion involves two steps which require some effort:
(i) Inequalities of the type (3.1.4) and (3.1.5) must be translated into inequalities on the function . (ii) Results obtained about the zeros of , lying in the unit disk, must be translated into results about the zeros of .
Regarding step (i), it turns out that inequalities of the form (3.1.5), and generalizations of it, are converted into inequalities of the form
where and the parameters in (3.1.6) are determined by those appearing in the inequality bounding and by properties of the conformal mapping . Note that this inequality restricts the growth of as differently according to whether or not approaches one of the ‘special’ points . Since functions obeying (3.1.6) play an important role in our work, it is convenient to have a special notation for this class of functions. First, let us set
Let . For let and