On canonical factorization of rational matrix functions

This paper concerns the problem of canonical factorization of a rational matrix functionW() which is analytic but may benot invertible at infinity. The factors are obtained explicitly in terms of the realization of the original matrix function. The cases of symmetric factorization for selfadjoint and positive rational matrix functions are considered separately.     

Left versus right canonical Wiener-Hopf factorization and realization

For an arbitrary rational matrix function, not necessarily analytic at infinity, the existence of a right canonical Wiener-Hopf factorization is characterized in terms of a left canonical Wiener-Hopf factorization. Formulas for the factors in a right factorization are given in terms of the formulas for the factors in a given left factorization. All formulas are based on a special representation of a rational matrix function involving a quintet of matrices.     

Generalized hermitian operators

In this paper, the concept of generalized hermitian operators defined on a complex Hilbert space is introduced. It is shown that the spectrums and the Fredholm fields of generalized hermitian operators are both symmetric with respect to the real axis. Some other results on generalized hermitian operators are obtained.     

Explicit Wiener-Hopf factorization for certain non-rational matrix functions

Explicit Wiener-Hopf factorizations are obtained for a certain class of nonrational 2×2 matrix functions that are related to the scattering matrices for the 1-D Schrödinger equation. The diagonal elements coincide and are meromorphic and nonzero in the upper-half complex plane and either they vanish linearly at the origin or they do not vanish. The most conspicuous nonrationality consists of imaginary exponential factors in the off-diagonal elements.     

Inertia theorems for operator pencils and applications

In this paper we obtain general infinite dimensional inertia theorems for linear pencils in Hilbert space which cover previously known results for the finite dimensional case and for block weighted shifts. Connections with definite subspaces for contractions in spaces with indefinite metric are discussed.     

A method of explicit factorization of matrix functions and applications

A method of explicit factorization of matrix functions of second order is proposed. The method consists of reduction of this problem to two scalar barrier problems and a finite system of linear equations. Applications to various classes of singular integral equations and equations with Toeplitz and Hankel matrices are given.     

Left and right factorizations of rational matrix functions

The right partial indices of the symbol are described in terms of realizations of factors of the left Wiener-Hopf canonical factorization of the same symbol. The dual results are also stated. Application to Wiener-Hopf equations is considered.     

Convolution equations on finite intervals and factorization of matrix functions

Systems of convolution equations on a finite interval are reduced to the problem of canonical factorization of unimodular matrix-valued functions. The discrete version is considered separately.This research was partially supported by the Israel Science Foundation funded by the Israel Academy of Sciences and Humanities.     

Factorization of matrix functions and characteristic properties of the circle

In the present paper we discuss two problems on factorizations of matrix-valued functions with respect to a simple closed rectifiable curve . These two problems are related and we show that in both of them circular contours play a remarkable role.     

Factorizations of and extensions to J-unitary rational matrix functions on the unit circle

This paper concerns two topics: (1) minimal factorizations in the class ofJ-unitary rational matrix functions on the unit circle and (2) completions of contractive rational matrix functions on the unit circle to two by two block unitary rational matrix functions which do not increase the McMillan degree. The results are given in terms of a special realization which does not require any additional properties at zero and at infinity. The unitary completion result may be viewed as a generalization of Darlington synthesis.     

On some properties of factorization indices

Various kinds of factorization indices are considered (right partial indices, left partial indices, Birkhoff indices), and some connections between them are described. We solve also the problem on the relation between the partial indices of two matrix functions and of their product.Dedicated to the memory of Mark Grigorievich KreinThis research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

Quasicomplete factorizations of rational matrix functions

It is shown that within the class ofn×n rational matrix functions which are analytic at infinity with valueW()=I _n, any rational matrix functionW is the productW=W ₁...W _p of rational matrix functionsW ₁,...,W _p of McMillan degree one. Furthermore, such a factorization can be established with a number of factors not exceeding 2(W)–1, where (W) denotes the McMillan degree ofW.     

Generalized factorization for a class ofn×n matrix functions — Partial indices and explicit formulas

The generalized factorization of a class of continuous non-rationaln×n matrix-functions is studied. The partial indices are determined and, in the case of existence of a canonical factorization, explicit formulas for the factors are obtained.     

Local minimal factorizations of rational matrix functions in terms of null and pole data: Formulas for factors

It is known that local minimal factorizations of a rational matrix function can be described in terms of local null and pole data (expressed in the form of left null-pole triples and their corestrictions) of this function. In this paper we give formulas for the factors in a local minimal factorization that corresponds to a given corestriction of the left null-pole triple.The first version of this paper was written while the second author visited the College of William and Mary.Partially supported by the NSF grant DMS-8802836 and by the Binational United States-Israel Foundation grant.     

The Schur Transformation for Generalized Nevanlinna Functions: Interpolation and Self-Adjoint Operator Realizations

The Schur transformation for generalized Nevanlinna functions has been defined and applied in [2]. In this paper we discuss its relation to a basic interpolation problem and study its effect on the minimal self-adjoint operator (or relation) realization of a generalized Nevanlinna function. D. Alpay acknowledges with thanks the Earl Katz family for endowing the chair which supported this research and the Netherlands Organization for Scientific Research, NWO (grant B 61-524). The research of A. Dijksma and H. Langer was partly supported by the Center for Advanced Studies in Mathematics, CASM, of the Department of Mathematics of Ben-Gurion University, that of H. Langer also by the Austrian Science Fund, Project P15540-N05. Received: September 25, 2006. Accepted: October 11, 2006.     

Some properties of the kernel and the cokernel of Toeplitz operators with matrix symbols

The relations between the kernels, as well as the cokernels, of Toeplitz operators are studied in connection with certain relations between their symbols. These results are used to obtain some Fredholm type properties for operators with 2×2 symbols, whose determinant admits a bounded Wiener-Hopf factorization.