Tools for the eigenvalue distribution in a non-Hermitian setting

Under mild trace norm assumptions on the perturbing sequence, we extend a recent perturbation result based on a theorem by Mirsky. The analysis concerns the eigenvalue distribution and localization of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices. Some examples of application are considered, ranging from the product of Toeplitz sequences to the approximation of PDEs with given boundary conditions. A final discussion on open questions and further lines of research ends the note.     

On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials

The present paper is devoted to the study of spectral properties of random Schrödinger operators. Using a finite section method for Toeplitz matrices, we prove a Wegner estimate for some alloy type models where the single site potential is allowed to change sign. The results apply to the corresponding discrete model, too. In certain disorder regimes we are able to prove the Lipschitz continuity of the integrated density of states and/or localization near spectral edges.     

Infinite analogues of block toeplitz matrices and related orthogonal functions

This paper concerns a class of infinite block matrices that are analogous to finite block Toeplitz matrices. Also studied are corresponding matrix-valued functions that are orthogonal for a matrixvalued inner product. An appendix presents basic results on orthogonalization in a Hilbert module.     

Approximate Factorization in Generalized Hardy Spaces

In this paper we establish a connection between the approximate factorization property appearing in the theory of dual algebras and the spectral inclusion property for a class of Toeplitz operators on Hilbert spaces of vector valued square integrable functions. As an application, it follows that a wide range of dual algebras of subnormal Toeplitz operators on various Hardy spaces associated to function algebras have property (A ₁(1)). It is also proved that the dual algebra generated by a spherical isometry (with a possibly infinite number of components) has the same property. One particular application is given to the existence of unimodular functions sitting in cyclic invariant subspaces of weak* Dirichlet algebras. Moreover, by this method we provide a unified approach to several Toeplitz spectral inclusion theorems. Research partially supported by grant CNCSIS GR202/2006 (cod 813).     

On a Toeplitz determinant identity of Borodin and Okounkov

In this note we give two other proofs of an identity of A. Borodin and A. Okounkov which expresses a Toeplitz determinant in terms of the Fredholm determinant of a product of two Hankel operators. The second of these proofs yields a generalization of the identity to the case of block Toeplitz determinants.Supported by National Science Foundation grant DMS-9970879.Supported by National Science Foundation grant DMS-9732687.     

Products of Toeplitz operators

An aspect of the theory of Toeplitz operators on generalised Hardy spaces is considered, namely, a necessary and sufficient condition on the symbols to ensure that the product of two Toeplitz operators is itself a Toeplitz operator. The answer to this question draws on many deep results of the theory of generalised Hardy spaces.     

Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture

We prove two-weight norm inequalities for Calderón-Zygmund singular integrals that are sharp for the Hilbert transform and for the Riesz transforms. In addition, we give results for the dyadic square function and for commutators of singular integrals. As an application we give new results for the Sarason conjecture on the product of unbounded Toeplitz operators on Hardy spaces.     

Products of Toeplitz Operators on the Bergman Space

In 1962 Brown and Halmos gave simple conditions for the product of two Toeplitz operators on Hardy space to be equal to a Toeplitz operator. Recently, Ahern and Cucković showed that a similar result holds for Toeplitz operators with bounded harmonic symbols on Bergman space. For general symbols, the situation is much more complicated. We give necessary and sufficient conditions for the product to be a Toeplitz operator (Theorem 6.1), an explicit formula for the symbol of the product in certain cases (Theorem 6.4), and then show that almost anything can happen (Theorem 6.7).     

The eigenvalue distribution of products of Toeplitz matrices - Clustering and attraction

We use a recent result concerning the eigenvalues of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product h, is described by the function h in the “Szegö way”. Then, using Mergelyan’s theorem, we extend the result to the more general case where h belongs to the Tilli class. The same technique gives us the analogous result for sequences belonging to the algebra generated by Toeplitz sequences, if the symbols associated with the sequences are bounded and the global symbol h belongs to the Tilli class. A generalization to the case of multilevel matrix-valued symbols and a study of the case of Laurent polynomials not necessarily belonging to the Tilli class are also given.     

The Constants in the Asymptotic Formulas by Rambour and Seghier for Inverses of Toeplitz Matrices

Rambour and Seghier established two deep results on the first order asymptotics of the entries of the inverses of certain Toeplitz matrices. The determination of the constants in the leading terms of the asymptotics turns out to be nontrivial. We here show how these constants can be obtained from a well known identity by Duduchava and Roch.Submitted: October 20, 2003     

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.     

Toeplitz operators with frequency modulated semi-almost periodic symbols

It is well known that amplitude modulation does not affect Fredholmness of Toeplitz operators. The same is true for frequency modulation provided the symbol of the operator is piecewise continuous. In this article, it is shown that frequency modulation can destroy Fredholmness for Toeplitz operators with almost periodic symbols; the corresponding example is based on the observation that certain almost periodic functions become semi-almost periodic functions after appropriate frequency modulation. Moreover, this article contains several results that can be employed in order to decide whether a Toeplitz operator with a frequency modulated semi-almost periodic symbol is Fredholm.     

On explicit factorization and applications

This paper is devoted to two topics connected with factorization of triangular 2 by 2 matrix functions. The first application is concerned with explicit factorization of a class of matrices of Daniel-Khrapkov type and the second is related to inversion of finite Toeplitz matrices. In the first section we present the scheme of factorization of triangular 2 by 2 matrix functions.     

Inversion formulas for infinite generalized Toeplitz matrices

Inversion formulas are obtained for a certain class of infinite matrices that possess displacement structure similar to that of finite block Toeplitz matrices. Consequences are symmetric inversion formulas for matrix-valued singular integral operators and infinite Toeplitz plus Hankel matrices.     

Automorphisms on the Toeplitz Algebra with Piecewise Continuous Symbol on the Unit Ball

In this paper, we obtain a Fredholm index formula for Toeplitz operators whose symbols are certain piecewise continuous function matrices on the unit ball. Moreover, using this formula, we discuss the automorphisms on the corresponding Toeplitz algebra     

INVERSION OF TOEPLITZ OPERATORS WITH RATIONAL SYMBOLS VIA FACTORIZATION AND REALIZATION

Abstract

Semi-infinite block Toeplitz operators with rational matrix symbols are inverted explicitly by employing the factorization method. All formulas are based on a special representation of the symbol involving a quintet of matrices, which differs from the ones that have been used earlier.     

Circulants,displacements and decompositions of matrices

In this paper are suggested new formulas for representation of matrices and their inverses in the form of sums of products of factor circulants, which are based on the analysis of the factor cyclic displacement of matrices. The results in applications to Toeplitz matrices generalized the Gohberg-Semencul, Ben-Artzi-Shalom and Heinig-Rost formulas and are useful for complexity analysis.     

Toeplitz operators on the Fock space: Radial component effects

The paper is devoted to the study of specific properties of Toeplitz operators with (unbounded, in general) radial symbolsa=a(r). Boundedness and compactness conditions, as well as examples, are given. It turns out that there exist non-zero symbols which generate zero Toeplitz operators. We characterize such symbols, as well as the class of symbols for whichT _a =0 impliesa(r)=0 a.e. For each compact setM there exists a Toeplitz operatorT _a such that spT _a =ess-spT _a =M. We show that the set of symbols which generate bounded Toeplitz operators no longer forms an algebra under pointwise multiplication.Besides the algebra of Toeplitz operators we consider the algebra of Weyl pseudodifferential operators obtained from Toeplitz ones by means of the Bargmann transform. Rewriting our Toeplitz and Weyl pseudodifferential operators in terms of the Wick symbols we come to their spectral decompositions.This work was partially supported by CONACYT Project 27934-E, México.The first author acknowledges the RFFI Grant 98-01-01023, Russia.