Formulas for the Inverses of Toeplitz Matrices with Polynomially Singular Symbols

We consider large finite Toeplitz matrices with symbols of the form (1– cos )^p f() where p is a natural number and f is a sufficiently smooth positive function. By employing techniques based on the use of predictor polynomials, we derive exact and asymptotic formulas for the entries of the inverses of these matrices. We show in particular that asymptotically the inverse matrix mimics the Green kernel of a boundary value problem for the differential operator Submitted: June 20, 2003     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

Normality of Products of <Emphasis Type="Italic">p</Emphasis>-Hyponormal Operators and Their Aluthge Transforms

Let B(H) denote the algebra of operators on a complex separable Hilbert space H, and let A $\in$ B(H) have the polar decomposition A = U|A|. The Aluthge transform is defined to be the operator . We say that A $\in$ B(H) is p-hyponormal, . Let . Given p-hyponormal , such that AB is compact, this note considers the relationship between denotes an enumeration in decreasing order repeated according to multiplicity of the eigenvalues of the compact operator T (respectively, singular values of the compact operator T). It is proved that is bounded above by and below by for all j = 1, 2, . . . and that if also is normal, then there exists a unitary U¹ such that for all j = 1, 2, . . ..     

Beurling’s Phenomenon in Two Variables

In the Hardy space over the unit disk H²(D), every shift-invariant subspace M is of the form H²(D)) for some inner function by Beurlings theorem, and the reproducing kernel of M is . The fact that is inner implies that is subharmonic and has boundary value 1 almost everywhere on T. In the two variable space H²(D²), things are far more complicated and there is no similar characterization of invariant subspaces M in terms of inner functions. However, we will show in this paper an analogous phenomenon in terms of reproducing kernels, namely, is subharmonic and has boundary value 1 almost everywhere on T². The proof uses an index theorem obtained recently.     

Bounded Reflexivity of Operator Spaces

We introduce a concept bounded reflexivity for a subspace of operators on a normed space. We explore the properties of bounded reflexivity, study the similarities and differences between bounded reflexivity and the usual reflexivity for a subspace of operators. As applications of bounded reflexivity, we give alternative proofs of some well known results about positivity and complete positivity of elementary operators.     

Decoupling of Proper Transfer Functions

This paper studies the state space and feedback aspects of linear system decoupling. Given a minimal realization for a proper transfer function W (s), a general procedure is given for the parametrization of all the minimal decouplings of W (s) into two proper subsystems. This completes and unifies known results on factorization and cascade decomposition.     

Solution of the Truncated Parabolic Moment Problem

Given real numbers with ₀₀ >0 , the truncated parabolic moment problem for entails finding necessary and sufficient conditions for the existence of a positive Borel measure , supported in the parabola p(x, y) = 0, such that We prove that admits a representing measure (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated and has a column relation p(X, Y) = 0, and the algebraic variety () associated to satisfies card In this case, admits a rank -atomic (minimal) representing measure.Submitted: August 25, 2003     

Hyponormal Operators with Rank One Self-Commutator and Quadrature Domains

This paper studies the class of pure hyponormal operators with rank one self-commutator satisfying the condition that their spectra are quadrature domains.     

Perron Conditions for Pointwise and Global Exponential Dichotomy of Linear Skew-Product Flows

The aim of this paper is to give necessary and sufficient conditions for pointwise and uniform exponential dichotomy of linear skew-product flows. We shall obtain that the pointwise exponential dichotomy of a linear skew-product flow is equivalent to the pointwise admissibility of the pair As a consequence, we prove that a linear skew-product flow on is uniformly exponentially dichotomic if and only if the pair is uniformly admissible for .     

A Note on *-paranormal Operators

In this note we show that if either T or T^* is totally *-paranormal then Weyls theorem holds for f(T) for every f , and also a-Weyls theorem holds for f(T) if T is totally *-paranormal. We prove that if either T or T^* is *-paranormal then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.