J-pseudo-spectral andJ-inner-pseudo-outer factorizations for matrix polynomials

For a comonic polynomialL() and a selfadjoint invertible matrixJ the following two factorization problems are considered: firstly, we parametrize all comonic polynomialsR() such that . Secondly, if it exists, we give theJ-innerpseudo-outer factorizationL()=()R(), where () isJ-inner andR() is a comonic pseudo-outer polynomial. We shall also consider these problems with additional restrictions on the pole structure and/or zero structure ofR(). The analysis of these problems is based on the solution of a general inverse spectral problem for rational matrix functions, which consists of finding the set of rational matrix functions for which two given pairs are extensions of their pole and zero pair, respectively.The work of this author was supported by the USA-Israel Binational Science Foundation (BSF) Grant no. 9400271.     

Block toeplitz operators with rational symbol analytic at infinity

This paper concerns the problem of explicit inversion of a block Toeplitz operator with rational and analytic at infinity symbol. The necessary and sufficient conditions for the invertibility and explicit formulas for the inverse are given in terms of the realization of the symbol.     

An algebra of integral operators with fixed singularities in kernels

We continue the study of algebras generated by the Cauchy singular integral operator and integral operators with fixed singularities on the unit interval, started in R.Duduchava, E.Shargorodsky, 1990. Such algebras emerge when one considers singular integral operators with complex conjugation on curves with cusps. As one of possible applications of the obtained results we find an explicit formula for the local norms of the Cauchy singular integral operator on the Lebesgue spaceL ₂ (, ), where is a curve with cusps of arbitrary order and is a power weight. For curves with angles and cusps of order 1 the formula was already known (see R.Avedanio, N.Krupnik, 1988 and R.Duduchava, N.Krupnik, 1995). Dedicated to Professor Israel Gohberg on the occasion of his 70-th birthday Supported by EPSRC grant GR/K01001     

Minimal Nonsquare <Emphasis Type="Italic">J</Emphasis>-Spectral Factorization,Generalized Bezoutians and Common Zeros for Rational Matrix Functions

The problem that we solve in this paper is to find (square or nonsquare) minimal J-spectral factors of a rational matrix function with constant signature. Explicit formulas for these J-spectral factors are given in terms of a solution of a particular algebraic Riccati equation. Also, we discuss the common zero structure of rational matrix functions that arise from the analysis of nonsquare J-spectral factors. This zero structure is obtained in terms of the kernel of a generalized Bezoutian.     

Tangential interpolation with symmetries and two-point interpolation problem for matrix-valued H2-functions

We describe all solutions of the two-sided tangential interpolation problem in the class of matrix-valued Hardy functions when symmetries are added: these symmetries are defined in terms of involutions ofH ₂. The obtained results are applied to a one-sided two-points tangential interpolation for matrix functions.The research of this author is partially supported by the NSF Grant DMS 9500924 and by the Binational United States-Israel Foundation Grant 9400271.     

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.     

Decomposition of a scalar operator into a product of unitary operators with two points in spectrum

We consider products of unitary operators with at most two points in their spectra, 1 and e^iα. We prove that the scalar operator e^iγI is a product of k such operators if α(1+1/(k-3))?γ?α(k-1-1/(k-3)) for k?5. Also we prove that for e^iα≠-1, only a countable number of scalar operators can be decomposed in a product of four operators from the mentioned class. As a corollary we show that every unitary operator on an infinite-dimensional space is a product of finitely many such operators.     

Toeplitz and singular integral operators on Carleson curves with logarithmic whirl points

We consider Toeplitz operators with piecewise continuous symbols and singular integral operators with piecewise continuous coefficients onL ^p (,w) where 1<p<,w is a Muckenhoupt weight and belongs to a large class of Carleson curves. This class includes curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. Our main result says that the essential spectrum of a Toeplitz operator is obtained from the essential range of its symbol by joining the endpoints of each jump by a certain spiralic horn, which may degenerate to a usual horn, a logarithmic spiral, a circular arc or a line segment if the curve and the weightw behave sufficiently well at the point where the symbol has a jump. This result implies a symbol calculus for the closed algebra of singular integral operators with piecewise continuous coefficients onL ^p (,w).Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer of the Krupp Foundation.     

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.     

Applications of Hilbert’s projective metric to a class of positive nonlinear operators

The purpose of this paper is to study the eigenvalue problems for a class of positive nonlinear operators. Using projective metric techniques and the contraction mapping principle, we establish existence, uniqueness and continuity results for positive eigensolutions of a particular type of positive nonlinear operator. In addition, we prove the existence of a unique fixed point of the operator with explicit norm-estimates. Applications to nonlinear systems of equations and to matrix equations are considered.     

Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces

The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some disintegration condition which combines properties of spaces and curves, the Boyd and spirality indices.We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic leaves) between the endpoints of jumps.These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra generated by SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIO's from in terms of their symbols.     

Analytic equivalence and similarity of operators

The analytic equivalence of two operators is a generalization of similarity. We prove that under some conditions the analytic equivalence between two Hilbert space operatorsT andR implies the similarity of their restrictions on generalized ranges. We also prove that, in certain cases, the similarity ofT to a contraction implies that ofR. An improvement of a well-known criterion of similarity to an isometry due to Sz.-Nagy is given and an extension of a result of Apostol is obtained.     

An equational approach to products of relatively regular operators

A bounded linear operatorT on Banach spaceX is called relatively regular if its nullspaceN(T) and rangeR(T) are closed complemented subspaces ofX. It is known that the product of two relatively regular operators is not necessarily relatively regular. This paper shows how to find conditions, more general than those previously known, to ensure that two relatively regular operators have relatively regular product.This work was supported in part by NRC Operating Grant A3985 and Canada Council Leave Fellowship.     

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.     

Factorization of staircase class operators in Hilbert space

A class of linear bounded staircase operators (H, G spaces) defined by (1) with two infinite sequences of orthogonal decompositions ofH and chain property (2) is considered. Necessary and sufficient conditions for the factorizationZ=XY are obtained, whereX, Y are block-diagonal, bounded, andY has a bounded inverse. All the pairs (X, Y) are explicitly constructed. These conditions are specialized for finite and infinite dimensions of the blocks ofX, Y and for differentX, Y. A direct application to bitriangular and biquasitriangular operators is indicated.     

Eigenvalues of integral operators onL 2(I) given by analytic kernels

Let {_n} _n=0 be the eigenvalue sequence of a symmetric Hilbert-Schmidt operator onL ₂(I). WhenI is an open interval, a necessary condition for {_n} _n=0 to be in the sequence space is obtained. WhenI is a closed bounded interval, sufficient conditions for {_n} _n=0 to be in the sequence space ^– are obtained.     

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.