The Kreîn-Langer problem for Hilbert space operator valued functions on the band

We deal with the Kreîn-Langer problem for -valued functions on the band (–2a, 2a)×, where is the algebra of continuous linear operators on a Hilbert space ,a a finite positive number and a topological Abelian group. We show that every weakly continuous -indefinite function admits a strongly continuous -indefinite continuation to × with the same indefiniteness index . We give a parametrization of the extensions in terms of operator-valued Schur functions.     

Extension of Locally Defined Indefinite Functions on Ordered Groups

We give a definition of -indefinite function of archimedean type, on an interval of an ordered group with an archimedean point. We say that has the indefinite extension property if every continuous -indefinite function of archimedean type, on an interval of , can be extended to a continuous -indefinite function on the whole group .We show that if a group is semi-archimedean and it has the indefinite extension property, then with the lexicographic order and the product topology has the indefinite extension property. As a corollary it is obtained that the groups and with the lexicographic order and the usual topologies, have the indefinite extension property.To Professor José R. León, for his kind encouragement.Both authors were supported in part by the CDCH of the Univ. Central de Venezuela and by CONICIT grant G-97000668. Both authors were visitors at IVIC during the realization of this paper.Submitted: August 8, 2002 Revised: January 30, 2003     

Subnormality and 2-hyponormality for Toeplitz operators

In this article we provide an example of a Toeplitz operator which is 2-hyponormal but not subnormal, and we consider 2-hyponormal Toeplitz operators with finite rank self-commutators.Supported by NSF research grant DMS-9800931.Supported by KOSEF research project No. R01-2000-00003.     

Logarithms and imaginary powers of closed linear operators

The imaginary powersA ^it of a closed linear operatorA, with inverse, in a Banach spaceX are considered as aC ₀-group {exp(itlogA);t R} of bounded linear operators onX, with generatori logA. Here logA is defined as the closure of log(1+A) – log(1+A ^–1). LetA be a linearm-sectorial operator of typeS(tan ), 0(/2), in a Hilbert spaceX. That is, |Im(Au, u)| (tan )Re(Au, u) foru D(A). Then ±ilog(1+A) ism-accretive inX andilog(1+A) is the generator of aC ₀-group {(1+A) ^it ;t R} of bounded imaginary powers, satisfying the estimate (1+A) ^it exp(|t|),t R. In particular, ifA is invertible, then ±ilogA ism-accretive inX, where logA is exactly given by logA=log(1+A)–log(1+A ^–1), and {A ^it;t R} forms aC ₀-group onX, with the estimate A ^it exp(|t|),t R. This yields a slight improvement of the Heinz-Kato inequality.     

On unbounded hyponormal operators II

The paper deals with unbounded hyponormal operators. Among others it is proved that any closed hyponormal operator with spectrum contained in a parabola generates a cosine function.     

Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

We consider hypercyclic composition operators on which can be obtained from the translation operator using polynomial automorphisms of . In particular we show that if C _S is a hypercyclic operator for an affine automorphism S on , then for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on ℓ₁. Received: 8 June 2006 Revised: 26 September 2006     

Zero-pole interpolation for meromophic matrix functions on an algebraic curve and transfer functions of 2D systems

We formulate and solve the problem of constructing a meromorphic bundle map over a compact Riemann surface X having a prescribed zero-pole structure (including directional information). The output bundle together with the zero-pole data is prespecified while the input bundle and the bundle map are to be determined. The Riemann surface X is assumed to be (birationally) embedded as an irreducible algebraic curve in ² and both input and output bundles are assumed to be equal to the kernels of determinantal representations for X. In this setting the solution can be found as the joint transfer function of a Livsic-Kravitsky two-operator commutative vessel (2D input-output dynamical system). Also developed is the basic theory of two-operator commutative vessels and the correct analogue of the transfer function for such a system (a meromorphic bundle map between input and output bundles defined over an algebraic curve associated with the vessel) together with a state space realization, a Mittag-Leffler type interpolation theorem and the state space similarity theorem for such bundle mappings. A more abstract version of the zero-pole interpolation problem is also presented.     

Polynomial sequences of bounded operators

The basic results of spectral theory are obtained using the sequence of powers of a bounded linear operator T,T²,…,Tⁿ,…. In this paper, we replace the powers Tⁿ by certain polynomials p_n(T), and make use of special properties of the polynomial sequence to derive some new results concerning operators. For example, using an arbitrary polynomial sequence , we obtain “binomial” spectral radii and semidistances, which reduce, in the case of the sequence of powers, to the usual spectral radius and semidistance.     

Equivalence between the dilation and lifting properties of an ordered group through multiplicative families of isometries. A version of the commutant lifting theorem on some lexicographic groups

Let be a locally compact abelian ordered group. has the dilation property if a special extension of the Naimark dilation theorem holds for and it has the commutant lifting property if a natural extension of the Sz.-Nagy — Foias commutant lifting theorem holds for .We prove that these two conditions are equivalent and we give another necessary and sufficient condition in terms of unitary extensions of multiplicative families of partial isometries.A version of the commutant lifting theorem is given for the groups ⁿ and × ⁿ with the lexicographic order and the natural topologies.Both authors were partially supported by the CDCH of the Universidad Central de Venezuela, and by CONICIT grant G-97000668.     

Joint hyponormality of composition operators with linear fractional symbols

We study joint hyponormality and joint subnormality of ofn-tuples of commuting composition operators with linear fractional symbols, acting on the Hardy spaceH ². We also consider subnormality ofn-tuples of adjoints of composition operators.     

Unimodular eigenvalues and invariant measures for linear operators

For a bounded linear operator in a complex separable Hilbert space we show that it accepts an invariant Borel probability measure of square integrable norm whose essential support spans the space, iff its eigenvectors with unimodular eigenvalues span the space.This research has been supported by the Economic Research Center (KOE) of the Athens University of Economics and Business.     

Extensions of bitriangular operators

We prove that every one-dimensional extension of a bitriangular operator has a cyclic commutant. We also prove that ifT is an extension of a bitriangular operator by an algebraic operator, then the weakly closed algebraW(T) generated byT has a separating vector.This work was partially supported by NSF Grant DMS-9401544.Participant, Workshop in Linear Analysis and Probability, Texas A&M University     

An operator version of the Newman-Shapiro isometry theorem

We establish an operator version of the Newman — Shapiro Isometry Theorem for operators satisfying generalized canonical commutation relations. An application to operator inequalities is also given.     

Geometric properties of positive definite matrices cone with respect to the Thompson metric

We shall discuss geometric properties of a quadrangle with parallelogramic properties in a convex cone of positive definite matrices with respect to Thompson metric.