Generalization of the Newman-Shapiro isometry theorem and Toeplitz operators

We give a generalization of the Newman-Shapiro Isometry Theorem to the case of Hilbert space-valued entire functions, which are square-summable with respect to the Gaussian measure on ⁿ , together with some applications in the theory of Toeplitz operators with operator-valued symbols. The study of various properties (such as density of domains, cores, closedness and boundedness from below) of these operators in illustrated with many relevant examples.Research supported by KBN under grant no. 2 P03A 041 10.     

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.     

Algebraic reflexivity of some subsets of the isometry group

Let X be a compact first countable space. In this paper we show that the set of isometries of C(X) that are involutions is algebraically reflexive. As a consequence of a recent work of Botelho and Jamison this leads to the conclusion that the set of generalized bi-circular projections on C(X) is also algebraically reflexive. We also consider these questions for the space C(X,E) where E is a uniformly convex Banach space.     

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     

Unitary extensions of two-parameter local semigroups of isometric operators and the Krein extension theorem

A notion of two-parameter local semigroups of isometric operators in Hilbert space is discussed. It is shown that under certain conditions such a semigroup can be extended to a strongly continuous two-parameter group of unitary operators in a larger Hilbert space. As an application a simple proof of the Eskin bidimensional version of the Krein extension theorem is given.     

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.     

Error estimate in operator norm for Trotter-Kato product formula

In the present paper we study the error estimate in the operator norm for the Trotter-Kato product formula, having an intention to apply the obtained result to Schrödinger semigroups with singular potentials.     

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 .     

Commutant lifting theorem and interpolation in discrete nest algebras

Ball in [Ba] showed that the commutant lifting theorem for the nest algebras due to Paulsen and Power gives a unified approach to a wide class of interpolation problems for nest algebras. By restricting our attention to the case when nest algebras associated with the problems are discrete we derive a variant of the commutant lifting theorem which avoids language of representation theory and which is sufficient to treat an analog of the generalized Schur-Nevannlinna-Pick (SNP) problem in the setting of upper triangular operators.     

Weyl's theorem and quasi-similarity

In this paper we give necessary and sufficient conditions of a finitely ascensive operator to obey Weyl's theorem and study Weyl's theorem for quasi-affine transforms. In particular, we give an improvement of the earlier result of Duggal forp-hyponormal operators.     

On pure subnormal operators with finite rank self-commutators and related operator tuples

In this paper, we study the model of a pure subnormal operator with finite rank self-commutator and of the relatedn-tuple of commuting linear bounded operators. We also give some applications of the model to the theory ofn-tuples of commuting operators with trace class self-commutators.This work is supported in part by a NSF grant no. DMS-9400766.     

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.     

On the decomposition theorem for meromorphic Fredholm resolvents

In this note results of B. Gramsch and W. Kaballo [8] on the decomposition of meromorphic (semi-) Fredholm resolvents are sharpened. A condition on an Orlicz function is given, under which the singular part in this decomposition can be chosen meromorphic inN , the ideal of -nuclear operators. Then the necessity of this condition is studied. Moreover, it is shown that for the rather steep Orlicz functions relevant to this question,N equalsS , the ideal of -approximable operators.Dedicated to Professor Albert Schneider on the occasion of his 60 ^th birthdayresearch supported by a grant from DAAD     

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.     

H-ULTRABORNOLOGICAL SPACES

Abstract

In this paper we define a special type of H-locally convex space—H-ultrabornological space -. We shall characterize those H-locally convex spaces in terms of linear maps and of inductive limits.     

On the Range of the Aluthge Transform

Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .     

On the Isolated Points of the Spectrum of Paranormal Operators

For paranormal operator T on a separable complex Hilbert space we show that (1) Weyl’s theorem holds for T, i.e., σ(T) \ w(T) = π₀₀(T) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T) is self-adjoint (i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ)^*.     

Multilinear Commutators of Singular Integrals with Non Doubling Measures

Let be a Radon measure on which may be non-doubling. The only condition that must satisfy is for all and for some fixed In this paper, under this assumption, the L^p()-boundedness (1 < p < ) and certain weak type endpoint estimate are established for multilinear commutators, which are generated by Calderón-Zygmund singular integrals with RBMO() functions or with functions for r 1, where is a space of Orlicz type satisfying that if r = 1 and if r > 1.     

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.     

Transferring subnormality of adjoint composition operators

For an analytic map of the unit disk into itself, the subnormality ofC ^* on the Hardy space implies its subnormality on the Bergman space.Supported in part by National Science Foundation Grant DMS 8910140.I would like to thank Tom Kriete for his helpful discussions and suggestions in connection with this paper.