On three Krein extension problems and some generalizations

Three basic extension problems which were initiated by M. G. Krein are discussed and further developed. Connections with interpolation problems in the Carathéodory class are explained. Some tangential and bitangential versions are considered. Full characterizations of the classes of resolvent matrices for these problems are given and formulas for the resolvent matrices of left tangential problems are obtained using reproducing kernel Hilbert space methods.Dedicated to the memory of M. G. Krein, a beacon for us both.The authors wish to acknowledge the partial support of the Israel-Ukraine Exchange Program. D. Z. Arov also wishes to thank the Weizmann Institute of Science for partial support and hospitality; H. Dym wishes to thank Renee and Jay Weiss for endowing the chair which supports his research.     

A Wiener-Young type theorem for dual semigroups

The purpose of this paper is to obtain extensions of the Wiener-Young theorem for strongly continuous semigroups of positive operators in Banach lattices.     

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.     

An irreducible semigroup of non-negative square-zero operators

We construct an irreducible multiplicative semigroup of non-negative square-zero operators acting onL ^p [0,1), for 1p<.The main idea for this paper was developed at the 2nd Linear Algebra Workshop at Bled, Slovenia, in June 1999.The work of the three Slovenian authors was supported by the Research Ministry of Slovenia.This author's work was supported by a Division grant from Colby College.     

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.     

The courant-fischer theorem and the spectrum of selfadjoint block band Toeplitz operators

We show that ifT(F) is a selfadjoint block Toeplitz operator generated by a trigonometric matrix polynomialF, then the spectrum ofT(F) as well as the limiting set (F) of the eigenvalues of the truncationsT _n (F) is the union of a finite collection of segments (the spectral range ofF) and at most a finite set of points for which we give an upper bound.     

Wold Decompositions and the Unitary Model for Bi-isometries

In this work we use the unitary model associated to a bi-isometry, which appears for the first time in [1] and was recently rediscovered in [2], to characterize the summands in the Wold decompositions of a bi-isometry and to give new proofs of such decomposition theorems.     

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.     

Common invariant subspaces for collections of operators

Let be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that is finitely quasinilpotent at a vectorx ₀X whenever for any finite subset of the joint spectral radius of atx ₀ is equal 0. If such collection contains a non-zero compact operator, then and its commutant have a common non-trivial invariant, subspace. If in addition, is a collection of positive operators on a Banach lattice, then has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia.     

The invertibility of the product of unbounded Toeplitz operators

In this note I give necessary and sufficient conditions on outer functionsf andg for the operator to be bounded and invertible on H². I also discuss the relationship of this question to two open questions in operator theory and weighted norm inequalities.     

Bounded characteristic functions and models for noncontractive sequences of operators

The Sz.-Nagy-FoiaŞ functional model for completely non-unitary contractions is extended to completely non-coisometric sequences of bounded operatorsT = (T₁,...,T _d) (d finite or infinite) on a Hilbert space, with bounded characteristic functions. For this class of sequences, it is shown that the characteristic function θ_T is a complete unitary invariant. We obtain, as the main result, necessary and sufficient conditions for a bounded multi-analytic operator on Fock spaces to coincide with the characteristic function associated with a completely non-coisometric sequence of bounded operators on a Hilbert space. Research supported in part by a COBASE grant from the National Research Council. The first author was partially supported by a grant from Ministerul Educaţiei Şi Cercetarii. The second author was partially supported by a National Science Foundation grant.     

Ultrapowers and semigroups of operators

We study two semigroups of operators between Banach spaces, related with the finite representability ofc ₀ and ℓ₁. We show that these semigroups are open, have nice duality properties and can be characterized in terms of compact perturbations, and in terms of the properties of their ultrapowers. We obtain analogous results for their associated dual semigroups. Supported in part by DGICYT Grant PB 94-1052 (Spain). Supported by a postdoctoral Grant of the Ministry of Spain for Education and Science     

The Weiss conjecture on admissibility of observation operators for contraction semigroups

We prove the conjecture of George Weiss for contraction semigroups on Hilbert spaces, giving a characterization of infinite-time admissible observation functionals for a contraction semigroup, namely that such a functionalC is infinite-time admissible if and only if there is anM>0 such that for alls in the open right half-plane. HereA denotes the infinitesimal generator of the semigroup. The result provides a simultaneous generalization of several celebrated results from the theory of Hardy spaces involving Carleson measures and Hankel operators.     

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     

Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functions

Sarason interpolation and Toeplitz corona problems are studied for almost periodic matrix functions. Recent results on almost periodic factorization and related generalized Toeplitz operators are the main tools in the study.Supported in part by NSF Grant DMS 9500912Supported in part by NATO Collaborative Research Grant 950332Supported by NSF Grant DMS 9500924     

Stability and ergodicity of dominated semigroups

This paper is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG