Linear algebra and matrix theory have long been fundamental tools in mathematical disciplines as well as fertile fields for research in their own right. In this book, and in the companion volume, Topics in Matrix Analysis, we present classical and recent results of matrix analysis that have proved to be important to applied mathematics. The book may be used as an undergraduate or graduate text and as a self-contained reference for a variety of audiences. We assume background equivalent to a one-semester elementary linear algebra course and knowledge of rudimentary analytical concepts. We begin with the notions of eigenvalues and eigenvectors; no prior knowledge of these concepts is assumed.

Facts about matrices, beyond those found in an elementary linear algebra course, are necessary to understand virtually any area of mathematical science, whether it be differential equations; probability and statistics; optimization; or applications in theoretical and applied economics, the engineering disciplines, or operations research, to name only a few. But until recently, much of the necessary material has occurred sporadically (or not at all) in the undergraduate and graduate curricula. As interest in applied mathematics has grown and more courses have been devoted to advanced matrix theory, the need for a text offering a broad selection of topics has become more apparent, as has the need for a modern reference on the subject.

There are several well-loved classics in matrix theory, but they are not well suited for general classroom use, nor for systematic individual study. A lack of problems, applications, and motivation; an inadequate index; and a dated approach are among the difficulties confronting readers of some traditional references.

One historical motivation for introducing the complex numbers C was that polynomials with real coefficients might not have real zeroes. For example, a calculation reveals that {1 + i, 1 − i} are zeroes of the polynomial p(t) = t2 – 2t + 2, which has no real zeroes. All zeroes of any polynomial with real coefficients are, however, contained in C. In fact, all zeroes of all polynomials with complex coefficients are in C. Thus, C is an algebraically closed field: There is no field F such that C is a subfield of F, and such that there is a polynomial with coefficients from C and with a zero in F that is not in C.

The fundamental theorem of algebra states that any polynomial p with complex coefficients and of degree at least 1 has at least one zero in C. Using synthetic division, if p(z) = 0, then t − z divides p(t); that is, p(t) = (t − z)q(t), in which q(t) is a polynomial with complex coefficients, whose degree is 1 smaller than that of p. The zeroes of p are z, together with the zeroes of q. The following theorem is a consequence of the fundamental theorem of algebra.

Theorem. A polynomial of degree n ≥ 1 with complex coefficients has, counting multiplicities, exactly n zeroes among the complex numbers.

The multiplicity of a zero z of a polynomial p is the largest integer k for which (t − z)k divides p(t). If a zero z has multiplicity k, then it is counted k times toward the number n of zeroes of p. It follows that a polynomial with complex coefficients may always be factored into a product of linear factors over the complex numbers.

Abstract Selfadjoint extensions of a closed symmetric operator in a Hilbert space with equal deficiency indices are described by means of ordinary boundary triplets. In certain problems the more general notion of a boundary triplet of bounded type is needed. It will be shown that such triplets correspond in a certain way with the, in general infinite dimensional, graph perturbations of selfadjoint operators or relations. However, when considering selfadjoint exit spaces extensions for a symmetric operator by means of the so-called Kreĭn's formula one meets the notion of boundary relation (even when the deficiency indices are finite). Whereas ordinary boundary triplets and boundary triplets of bounded type correspond to bounded unitary or unitary operators in a Kreĭn space, respectively, boundary relations correspond to general unitary relations in a Kreĭn space, which are not necessarily single-valued. It is shown that the study of isometric relations in a Kreĭn space has useful applications. This present overview of recent developments includes illustrations, for instance, by means of elliptic differential operators and Schrödinger operators with local point interactions.

Introduction

The extension theory of densely defined symmetric operators was developed in the 1930s by J. von Neumann. A complete description was given for all selfadjoint extensions in terms of the defect subspaces; see [von Neumann, 1932; Stone, 1932]. Then M.H. Stone suggested to J.W. Calkin to develop another approach based on the notion of “abstract boundary conditions”, which reduces the extension problem to a description of the hyper-maximal symmetric subspaces of some auxiliary Hilbert space.

The theory of unbounded operators, which dates back to the early 1930s, was developed by J. von Neumann and M.H. Stone. Of course the earlier work of H. Weyl on boundary eigenvalue problems and of T. Carleman on singular integral operators should be mentioned as stepping stones for the general abstract treatment. One of the underlying ideas was to put quantum mechanics on a rigorous mathematical foundation. J. von Neumann used a distinction between symmetric (Hermitian) and selfadjoint (hypermaximal) operators and referred to E. Schmidt for this. Once this distinction was made, it was natural to determine all selfadjoint extensions of a symmetric operator (necessarily with equal deficiency indices). J. von Neumann gave such a description for densely defined symmetric operators by means of his well-known formulas. This description requires the knowledge of the deficiency spaces of the symmetric operator. Another approach involving abstract boundary conditions was developed by J.W. Calkin in his 1937 Harvard doctoral dissertation, which was written under the direction of Stone, who suggested the topic. Unfortunately Calkin's work on boundary value problems did not receive the attention it deserved; probably because he never returned to it after his mathematical work related to World War II.

A revival of interest in applications of this approach to boundary value problems is due to M.G. Kreĭn, M.I. Vishik, M.S. Birman, and R. S. Phillips in the 1950s and, later, to G. Grubb, F.S. Rofe-Beketov, and M.L. Gorbachuk.

Abstract This is a short survey on the connection between general extension theories and the study of realizations of elliptic operators A on smooth domains in ℝn, n ≥ 2. The theory of pseudodifferential boundary problems has turned out to be very useful here, not only as a formulational framework, but also for the solution of specific questions. We recall some elements of that theory, and show its application in several cases (including new results), namely to the lower boundedness question, and the question of spectral asymptotics for differences between resolvents.

Introduction

The general theory of extensions of a symmetric operator (or a dual pair of operators) in a Hilbert space, originating in the mid-1900's, has been applied in numerous works to ordinary differential equations (ODE), and also in a (smaller) number of works to partial differential equations (PDE).

There is a marked difference between the two cases: In ODE, the playground for boundary conditions is usually finite-dimensional vector spaces, where linear conditions can be expressed by the help of matrices. Moreover, the domains of differential operators defined by closure in L2- based Hilbert spaces can usually all be expressed in terms of functions with the relevant number of absolutely continuous derivatives.

In contrast, boundary conditions for PDE (in space dimensions n ≥ 2) are prescribed on infinite-dimensional vector spaces. Moreover, the domains of differential operators in L2-based spaces will contain functions with distribution derivatives, not continuous and possibly highly irregular.

Abstract Making use of Naĭmark extensions of a symmetric operator arising from an indeterminate Hamburger moment sequence we manufacture a machinery for providing representing measures with the following properties

1. 1o the support of each of them is in arithmetic progression;

2. 2o the supports of all the measures together partition ℝ;

3. 3o none of them is N-extremal;

4. 4o all of them are of infinite order.

All this is based on and, in fact, is a kind of guide to [Cichoń, Stochel and Szafraniec, 2010].

Moment problems and their close relatives, orthogonal polynomials were pretty often treated by the same means, mostly continuous fractions. Then, starting from [Stone, 1932, Chapter X] operator theory was used to handle the problem, look at [Landau, 1980] and [Fuglede, 1983, especially p. 51] for further references as well as for a fairly decent introduction to the subject, in a contemporary language.

Sustaining this idea in what refers to the single real variable case, sooner or later one has to deal with a symmetric operator which, as a matter of course, has deficiency indices (0, 0) or (1, 1). The latter is of interest here as it corresponds to an indeterminate moment problem and the routine procedure is to pass to extensions in the same Hilbert space as they always have to exist. The other way, which seems to be much less exploited with some exceptions, like [Kreĭn and Krasnoselskiĭ, 1947; Gil de Lamadrid, 1971; Langer, 1976; Simon, 1998] and references therein, is to go for extensions beyond the space.

Abstract This chapter is an introduction to the basic theory of state/signal systems via boundary control theory. The ℒC-transmission line illustrates the new concepts. It is shown that every boundary triplet can be interpreted as an impedance representation of a conservative boundary control state/signal system.

Introduction

We discuss the connection between some basic notions of boundary control state/signal systems on one hand, and classical boundary triplets on the other hand. Boundary triplets and their generalizations have been extensively utilized in the theory of self-adjoint extensions of symmetric operators in Hilbert spaces, see e.g. [Gorbachuk and Gorbachuk, 1991; Derkach and Malamud, 1995; Behrndt and Langer, 2007], and the references therein.

The notions related to standard input/state/output boundary control systems are discussed in Section 4.2, where we also introduce the boundary control state/signal system. In Section 4.3 we briefly discuss the concept of conservativity in the state/signal framework and in Section 4.4 we illustrate the abstract concepts we have introduced using the example of a finite-length conservative ℒC-transmission line with distributed inductance and capacitance.

We conclude this chapter in Section 4.5, where we recall the definition of a boundary triplet for a symmetric operator and compare this object to a boundary control state/signal system. In particular, we show that every boundary triplet can be transformed into a conservative boundary control state/signal system in impedance form, but that the converse is not true. We make a few final remarks about common generalizations of boundary triplets, which leads over to Chapter 5, where we treat more general passive state/signal systems, not only conservative systems or systems of boundary-control type.

Abstract This chapter is a continuation and deepening of Chapter 4. In the present chapter the state/signal theory is extended beyond boundary control and beyond conservative systems. The main aim is to clarify the basic connections between the state/signal theory and that of (conservative) boundary relations. It is described how one can represent a state/signal system using input/state/output systems in different ways by making different choices of input signal and output signal. There is an “almost one-to-one” relationship between conservative state/signal systems and boundary relations, and this connection is used in order to introduce dynamics to a boundary relation. Consequently, a boundary relation is such a general object that it mathematically has rather little to do with boundary control. TheWeyl family and γ-field of a boundary relation are connected to the frequency-domain characteristics of a state/signal system.

Introduction

The theory of boundary relations has been developed by a number of authors in the framework of the theory of self-adjoint extensions of symmetric operators and relations in Hilbert spaces; see e.g. the recent articles [Derkach et al., 2006; Derkach, 2009; Derkach et al., 2009; Behrndt et al., 2009]. Boundary relations are described in detail in Chapter 7.

One way of introducing the notion of a state/signal (s/s) system is to start from an input/state/output (i/s/o) system. By a standard i/s/o system we mean a system of equations of the type

where ẋ stands for the time derivative of x.

Abstract This chapter is a survey of results related to the problem of a description of all maximal accretive extensions for a densely defined sectorial operator; the problem in more generality was originally posed by R. Phillips. We also treat maximal sectorial extensions. Our approach uses the concepts of boundary pairs and boundary triplets.

Introduction

A linear operator S in a complex Hilbert space h is called accretive [Kato, 1995] if Re (Su, u) ≥ 0 for all u ∈ dom S. An accretive operator S is called maximal accretive (m-accretive) if one of the following equivalent conditions is satisfied [Kato, 1995; Lyantse, 1954; Phillips, 1959a, b, 1969]:

• the operator S is closed and has no accretive extensions in h;

• ρ(S) ∩ ≠ 0, where π_ denotes the open left half-plane;

• the operator S is densely defined and closed, and S* is accretive;

• the operator −S generates a one-parameter contractive semigroup T(t) = exp(−tS), t ≥ 0.

The resolvent set ρ(S) of an m-accretive operator contains the open left half-plane and

The class of m-accretive operators plays an essential role in differential equations, scattering theory, stochastic processes, passive linear systems and, for instance, hydrodynamics. It should be noted that Phillips calls an operator A dissipative if −A is accretive. Nowadays the term dissipative is used for operators A with Im (Af, f) ≥ 0.

The Phillips problem is to find all m-accretive extensions of a densely defined accretive operator; cf. [Phillips, 1959a, b, 1969].