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.     

Conditions for eigenvectors of a selfadjoint operator-function to form a basis

Consider a functionL() defined on an interval of the real axis whose values are linear bounded selfadjoint operators in a Hilbert spaceH. A point ₀ and a vector ₀ H( ₀ 0) are called eigenvalue and eigenvector ofL() ifL() ifL(₀) ₀ = 0. Supposing that the functionL() can be represented as an absolutely convergent Fourier integral, the interval is sufficiently small and the derivative will be positive at some point, it has been proved that all the eigenvectors of the operator-functionL() corresponding to the eigenvalues from the interval form an unconditional basis in the subspace spanned by them.     

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.     

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.     

Strong monotonicity of operator functions

LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A ^t ≤B ^t for somet>1 if and only if the null space ofB−A is invariant forA.     

On Factorization of Trigonometric Polynomials

We give a new proof of the operator version of the Fejér-Riesz Theorem using only ideas from elementary operator theory. As an outcome, an algorithm for computing the outer polynomials that appear in the Fejér-Riesz factorization is obtained. The extremal case, where the outer factorization is also *-outer, is examined in greater detail. The connection to Aglers model theory for families of operators is considered, and a set of families lying between the numerical radius contractions and ordinary contractions is introduced. The methods are also applied to the factorization of multivariate operator-valued trigonometric polynomials, where it is shown that the factorable polynomials are dense, and in particular, strictly positive polynomials are factorable. These results are used to give results about factorization of operator valued polynomials over , in terms of rational functions with fixed denominators.     

The Mosaic and Principal Function of a Subnormal Operator

In this paper, we use the mosaic of a subnormal operator given by Daoxing Xia to give an alternate definition of the Pincus principal function for pure subnormal operators. This allows us to provide much simplified proofs of some of the basic properties of the principal function and of the Carey-Helton-Howe-Pincus Theorem in the subnormal case.     

Classes of Operator-Smooth Functions - II. Operator-Differentiable Functions

This paper studies the spaces of Gateaux and Frechet Operator Differentiable functions of a real variable and their link with the space of Operator Lipschitz functions. Apart from the standard operator norm on B(H), we consider a rich variety of spaces of Operator Differentiable and Operator Lipschitz functions with respect to symmetric operator norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of Operator Differentiable functions. We apply the obtained results to the study of the functions acting on the domains of closed *-derivations of C*-algebras and prove that Operator Differentiable functions act on all such domains.We also obtain the following modification of this result: any continuously differentiable, Operator Lipschitz function acts on the domains of all weakly closed *-derivations of C*-algebras.     

Bezout operators for analytic operator functions,I. A general concept of Bezout operator

The notion of a Bezout operator, previously known for some special classes of scalar entire functions and for matrix and operator polynomials, is introduced for general analytic operator functions. Our approach is based on representing the operator functions involved in realized form. Basic properties of Bezout operators are established and known Bezout operators are shown to be specific realizations of our general concept.The work of this author was supported by the United States-Israel Binational Science Foundation Grant 88-00304.     

Finite Laurent Developments and the Logarithmic Residue Theorem in the Real Non-analytic Case

This paper develops a general abstract non-holomorphic operator calculus under minimal regularity requirements on the family of operators through the concept of algebraic eigenvalue and the use of a, very recent, transversalization theory. Further, it analyzes under what conditions the inverse of a non-analytic family admits a finite Laurent development, and employs the new findings to calculate the multiplicity of a real non-analytic family through a logarithmic residue, so extending the applicability of the classical theory of I. C. Gohberg and coworkers. Applications to matrix families and Nonlinear Analysis are also explained.     

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.     

Eigenfunctions of the hyperbolic composition operator

Letu inH ² be zero at one of the fixed points of a hyperbolic Möbius transform of the unit diskD. We will show, under some additional conditions onu, that the doubly cyclic subspaceS _u =V _n=– C ⁿ u contains nonconstant eigenfunctions of the composition operatorC . This implies that the cyclic subspace generated byu is not minimal. If there is an infinite dimensional minimal invariant subspace ofC (which is equivalent to the existance of an operator with only trivial invariant subspaces), then it is generated by a function with singularities at the fixed points of .     

Spectral properties of a compactly perturbed linear span of projections

Spectral properties of the sum of a linear span of projections and a compact nonnegative operator are considered. In particular, we prove partial completeness results for certain parts of the system of eigenfunctions. The main tool is a transformation the original spectral problem to that of a monic weakly hyperbolic pencil.