M. A. Kaashoek

Unbounded Linear Operators

This chapter gives an introduction to the theory of unbounded linear operators between Banach spaces. The important notions of closed and closable operators and their conjugates are analyzed with much attention paid to ordinary and partial differential operators. In particular, maximal and minimal operators and the properties of their inverses are studied. The chapter is divided into 6 sections. The first two sections are devoted to the general theory, and the other four sections deal mainly with differential operators.

Metric constrained interpolation, commutant lifting, and systems

Part 1 Interpolation and time-invariant system: interpolation problems for time-valued functions proofs using the commutant lifting theorem time invariant systems central commutant lifting central state space solutions parametization of intertwinning and its applications applications to control systems. Part 2 Nonstationary interpolation and time-varying systems nonstationary interpolation theorems nonstationary systems and point evaluation reduction techniques - from nonstationary to stationary and vice versa proofs of the nonstationary interpolation theorems by reduction to the stationary case a general completion theorem applications of the three chains completion theorem to interpolation parameterization of all solutions of the three chains completion problem.

Time varying linear systems with boundary conditions and integral operators. I. The transfer operator and its properties

The present paper is the first of a series of papers in which time varying linear systems with boundary conditions and integral operators with semi-separable kernels are studied. The interplay between the systems and the integral operators is one of the main features. A general theory of systems with boundary conditions is developed which includes a detailed study of minimality and minimal factorization. This first part has mainly an introductory character. It contains for systems with boundary conditions a systematic analysis of the concepts of transfer operator, realization, similarity, cascade connection and factorization. For discrete systems an analogous theory is developed.

The band method for positive and strictly contractive extension problems: An alternative version and new applications

The band method for positive and strictly contractive extension problems is deduced from a new set of axioms. New applications concern extension problems for operator-valued functions in the Wiener class and for certain infinite operator matrices.

Minimality and realization of discrete time-varying systems

The minimality and realization theory is developed for discrete time-varying finite dimensional linear systems with time-varying state spaces. The results appear as a natural generalization of the corresponding theory for the time-independent case. Special attention is paid to periodical systems. The case when the state space dimensions do not change in time is re-examined.

Nevanlinna-Pick interpolation for time-varying input-output maps: the discrete case

This paper presents the conditions of solvability and describes all solutions of the matrix version of the Nevanlinna-Pick interpolation problem for time-varying input-output maps. The system theoretical point of view is employed systematically. The technique of solution generalizes the method for finding rational solutions of the time-invariant version of the problem which is based on reduction to a homogeneous interpolation problem.

Equivalence, linearization, and decomposition of holomorphic operator functions

This paper discusses the problem of classifying holomorphic operator functions up to equivalence. A survey is given in §1 of the main results about equivalence classes of holomorphic matrix functions and holomorphic Fredholm-operator functions. In §2, it is shown that given a holomorphic function A on a bounded domain Ω into a space of bounded linear operators between two Banach spaces, it is possible to extend the operators A(λ) (for each λ ϵ Ω) by an identity operator IZ in such a way that the extended operator function A(·) ⊕ IZ is equivalent on Ω to a linear function of λ, T − λI. Other versions of this “linearization by extension” are described, including the cases of entire functions and polynomials (where Ω = C). As an application of these results, we consider the operator function equation A2(λ) Z2(λ) + Z1(λ) A1(λ) = C(λ), λ ϵ Ω, (∗) and explicitly construct the solutions Z1 and Z2. The formulas for Z1 and Z2 seem to be new, even when A1, A2 and C are matrix polynomials. The existence of solutions of (∗) makes it possible to analyze an operator function A whose spectrum decomposes into pairwise disjoint compact subsets σ1, …, σn of Ω. In this case, a suitable extension of A is equivalent on Ω to a direct sum of operator functions, A1, …, An, such that the spectrum of Ai is σi (i = 1, …, n). In the final section of the paper, we discuss the relation between local and global equivalence on Ω, and show that there exist operator functions A and B which are locally equivalent on Ω, but admit no extensions (of the sort considered in this paper) which are globally equivalent on Ω.

Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators

Abstract For a certain class of matrix functions that are analytic on the real line but not at infinity an explicit method of factorization and of constructing inverse Fourier transforms is developed. This method is applied to invert Wiener-Hopf integral equations on the half line and the full line. The results obtained extend analogous results of the authors for rational matrix functions and for functions that are analytic on the real line and at infinity. The analysis is based on an infinite dimensional realization theorem which involves operators that are a direct sum of two infinitesimal generators of C 0 -semigroups of negative exponential type, one of which has support on the negative half line and the other on the positive half line. The latter operators are called exponentially dichotomous and the study of their properties forms an essential part of the paper.

Minimal Divisors of Rational Matrix Functions with Prescribed Zero and Pole Structure

Necessary and sufficient conditions are given in order that a rational matrix function is a minimal divisor of another one. These conditions are expressed in terms of zero and pole structure of the given functions. In connection with this a description is obtained of all rational matrix functions with prescribed zero and pole data.

Constructive methods of Wiener-Hopf factorization

I: Canonical and Minimal Factorization.- Editorial introduction.- Left Versus Right Canonical Factorization.- 1. Introduction.- 2. Left and right canonical Wiener-Hopf factorization.- 3. Application to singular integral operators.- 4. Spectral and antispectral factorization on the unit circle.- 5. Symmetrized left and right canonical spectral factorization on the imaginary axis.- References.- Wiener-Hopf Equations With Symbols Analytic In A Strip.- 0. Introduction.- I. Realization.- 1. Preliminaries.- 2. Realization triples.- 3. The realization theorem.- 4. Construction of realization triples.- 5. Basic properties of realization triples.- II. Applications.- 1. Inverse Fourier transforms.- 2. Coupling.- 3. Inversion and Fredholm properties.- 4. Canonical Wiener-Hopf factorization.- 5. The Riemann-Hilbert boundary value problem.- References.- On Toeplitz and Wiener-Hopf Operators with Contour-Wise Rational Matrix and Operator Symbols.- 0. Introduction.- 1. Indicator.- 2. Toeplitz operators on compounded contours.- 3. Proof of the main theorems.- 4. The barrier problem.- 5. Canonical factorization.- 6. Unbounded domains.- 7. The pair equation.- 8. Wiener-Hopf equation with two kernels.- 9. The discrete case.- References.- Canonical Pseudo-Spectral Factorization and Wiener-Hopf Integral Equations.- 0. Introduction.- 1. Canonical pseudo-spectral factorizations.- 2. Pseudo-?-spectral subspaces.- 3. Description of all canonical pseudo-?-spectral factorizations.- 4. Non-negative rational matrix functions.- 5. Wiener-Hopf integral equations of non-normal type.- 6. Pairs of function spaces of unique solvability.- References.- Minimal Factorization of Integral operators and Cascade Decompositions of Systems.- 0. Introduction.- I. Main results.- 1. Minimal representation and degree.- 2. Minimal factorization (1).- 3. Minimal factorization of Volterra integral operators (1).- 4. Stationary causal operators and transfer functions.- 5. SB-minimal factorization (1).- 6. SB-minimal factorization in the class (USB)..- 7. Analytic semi-separable kernels.- 8. LU- and UL-factorizations (1).- II. Cascade decomposition of systems.- 1. Preliminaries about systems with boundary conditions.- 2. Cascade decompositions.- 3. Decomposing projections.- 4. Main decomposition theorems.- 5. Proof of Theorem II.4.1.- 6. Proof of Theorem II.4.2.- 7. Proof of Theorem II.4.3.- 8. Decomposing projections for inverse systems..- III. Proofs of the main theorems.- 1. A factorization lemma.- 2. Minimal factorization (2).- 3. SB-minimal factorization (2).- 4. Proof of Theorem I.6.1.- 5. Minimal factorization of Volterra integral operators (2).- 6. Proof of Theorem I.4.1.- 7. A remark about minimal factorization and inversion.- 8. LU- and UL-f actorizations (2).- 9. Causal/anticausal decompositions.- References.- II: Non-Canonical Wiener-Hopf Factorization.- Editorial introduction.- Explicit Wiener-Hopf Factorization and Realization.- 0. Introduction.- 1. Preliminaries.- 1. Peliminaries about transfer functions.- 2. Preliminaries about Wiener-Hopf factorization.- 3. Reduction of factorization to nodes with centralized singularities.- II. Incoming characteristics.- 1. Incoming bases.- 2. Feedback operators related to incoming bases.- 3. Factorization with non-negative indices.- III. Outgoing characteristics.- 1. Outgoing bases.- 2. Output injection operators related to outgoing bases.- 3. Factorization with non-positive indices.- IV. Main results.- 1. Intertwining relations for incoming and outgoing data.- 2. Dilation to a node with centralized singularities.- 3. Main theorem and corollaries.- References,.- Invariants for Wiener-Hopf Equivalence of Analytic Operator Functions.- 1. Introduction and main result.- 2. Simple nodes with centralized singularities.- 3. Multiplication by plus and minus terms.- 4. Dilation.- 5. Spectral characteristics of transfer functions: outgoing spaces.- 6. Spectral characteristics of transfer functions: incoming spaces.- 7. Spectral characteristics and Wiener-Hopf equivalence.- References.- Multiplication by Diagonals and Reduction to Canonical Factorization.- 1. Introduction.- 2. Spectral pairs associated with products of nodes.- 3. Multiplication by diagonals.- References.- Symmetric Wiener-Hopf Factorization of Self-Adjoint Rational Matrix Functions and Realization.- 0. Introduction and summary.- 1. Introduction.- 2. Summary.- I. Wiener-Hopf factorization.- 1. Realizations with centralized singularities..- 2. Incoming data and related feedback operators.- 3. Outgoing data and related output injection operators.- 4. Dilation to realizations with centralized singularities.- 5. The final formulas.- II. Symmetric Wiener-Hopf factorization.- 1. Duality between incoming and outgoing operators.- 2. The basis in (C and duality between the feedback operators and the output injection operators.- 3. Proof of the main theorems.- References.