Additive Maps Preserving Local Spectrum

Let X be a complex Banach space, and let be the space of bounded operators on X. Given and x ∈ X, denote by σ_T (x) the local spectrum of T at x. We prove that if is an additive map such that

Meromorphic Factorization Revisited and Application to Some Groups of Matrix Functions

Some properties and applications of meromorphic factorization of matrix functions are studied. It is shown that a meromorphic factorization of a matrix function G allows one to characterize the kernel of the Toeplitz operator with symbol G without actually having to previously obtain a Wiener–Hopf factorization. A method to turn a meromorphic factorization into a Wiener–Hopf one which avoids having to factorize a rational matrix that appears, in general, when each meromorphic factor is treated separately, is also presented. The results are applied to some classes of matrix functions for which the existence of a canonical factorization is studied and the factors of a Wiener–Hopf factorization are explicitly determined. Submitted: April 15, 2007. Revised: October 26, 2007. Accepted: December 12, 2007.     

Operator Valued Functions with Pick Operators Having Negative Subspaces of Bounded Dimensions

Functions whose values are bounded linear Hilbert space operators (each operator may be defined on its own subspace of the ambient Hilbert space), the domain of definition is contained in the open unit disc, and having the following property κ, are studied. (κ): All Pick operators associated with the function have the dimensions of their spectral subspace corresponding to the negative part of the spectrum bounded above by a fixed nonnegative integer κ, and the bound κ is attained. No a priori hypotheses concerning regularity of the functions are assumed. A particular class of functions, called standard functions, is introduced, and the corresponding nonnegative integer κ is identified for standard functions. It is proved that every function with property (κ) can be extended to a standard function with property (κ), for the same κ. This result is interpreted as a result on interpolation. As an application, maximal (with respect to the extension relation) functions with the property κ, for a fixed κ, are studied in terms of standard functions. Received: August 5, 2007., Accepted: October 24, 2007.     

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Passive systems with and as an input and output space and as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system with is said to be quasi-selfadjoint if ran . The subclass of the Schur class is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass and the Q-function of T is given. Received: December 16, 2007., Accepted: March 4, 2008.     

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Passive linear systems τ = have their transfer function in the Schur class S . Using a parametrization of contractive block operators the transfer function is connected to the Sz.-Nagy–Foiaş characteristic function of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions and . The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Dedicated to Eduard Tsekanovskiĭ on the occasion of his seventieth birthday This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 212146, 117617) and the Dutch Organization for Scientific Research N.W.O. (B 61-553). Received: December 22, 2006. Revised: February 6, 2007.     

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

Quadratic Inequalities for Hilbert Space Operators

Properties of sets of solutions to inequalities of the form

Spectral inclusions for operators on Banach spaces

This article centers around the relation between the spectra of two Banach space operators that are linked by some intertwining condition such as quasi-similarity. Certain conditions from local spectral theory are shown to be both necessary and sufficient for these operators to have equal spectra, approximate point spectra, or surjectivity spectra. A key role is played by a localized version of Bishop’s classical property (β) and a related closed range condition. As an application to harmonic analysis, the measures on a locally compact abelian group that avoid the Wiener-Pitt phenomenon are characterized in terms of local spectral theory.     

Bounds on Variation of Spectral Subspaces under J-Self-adjoint Perturbations

Let A be a self-adjoint operator on a Hilbert space . Assume that the spectrum of A consists of two disjoint components σ₀ and σ₁. Let V be a bounded operator on , off-diagonal and J-self-adjoint with respect to the orthogonal decomposition where and are the spectral subspaces of A associated with the spectral sets σ₀ and σ₁, respectively. We find (optimal) conditions on V guaranteeing that the perturbed operator L = A + V is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on the variation of the spectral subspaces of A under the perturbation V. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a -symmetric perturbation is discussed. This work was supported by the Deutsche Forschungsgemeinschaft (DFG), the Heisenberg-Landau Program, and the Russian Foundation for Basic Research.     

Generalized Essential Norm of Weighted Composition Operators on some Uniform Algebras of Analytic Functions

We compute the essential norm of a composition operator relatively to the class of Dunford-Pettis operators or weakly compact operators, on some uniform algebras of analytic functions. Even in the context of H^∞ (resp. the disk algebra), this is new, as well for the polydisk algebras and the polyball algebras. This is a consequence of a general study of weighted composition operators.      

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let denote the algebra of all functions that are holomorphic. If is a subalgebra of , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra , the disk algebra and the Wiener algebra are all infinite. Submitted: October 10, 2007., Revised: January 11, 2008., Accepted: January 12, 2007.     

Fock Spaces Connected with Spherical Mean Operator and Associated Operators

We define and study the Fock space associated with the spherical mean operator. Next, we establish some results for the Segal-Bergmann transform for this space. Lastly, we prove some properties concerning Toeplitz operators on this space. Received: May 11, 2007. Revised: May 20, 2008. Accepted: May 23, 2008.     

Relations Between Two Operator Inequalities Motivated by the Theory of Operator Means

We shall show several results on operator inequalities motivated by the theory of operator means. As a consequence of our main result, we shall also obtain relations between two operator inequalities

A Remark on Support of the Principal Function for Class A Operators

Let be an invertible class A operator such that . Then we show that , where gT is the principal function of T. Moreover, we show that if T is pure, then .     

Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffer-type linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman.      

Test Function Criteria for Hankel Operators

It is well known that there are classes of test functions such that a Hankel operator is bounded if and only if it is bounded on those functions. Criteria are derived which determine whether a Hankel operator is compact or belongs to a particular Schatten class, in terms of its action on those test functions.     

Spectral Radius of Refinement and Subdivision Operators with Power Diagonal Dilations

Two types of estimate for the spectral radius of the multivariate refinement operator with power diagonal dilations are presented. One type contains multiplicator norm of number matrices generated by the symbol of the corresponding operator and by specific subsets of repeating fractions. These subsets are used together with the little Fermat theorem to establish estimates that comprise integrals over tori of various dimensions. Moreover, we note certain classes of symbols when the exact value of the spectral radius of refinement operator can be found. For the spectral radius of subdivision operators point value estimates are established. Submitted: April 25, 2007. Accepted: November 5, 2007.     

Additive maps preserving the minimum and surjectivity moduli of operators

Let be a complex Hilbert space and let be the algebra of all bounded linear operators on . We characterize additive maps from onto itself preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the maximum modulus of operators. Received: 15 July 2008