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.      

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

Examples of operators on <Emphasis Type="Italic">C</Emphasis>[0, 1] distinguishing certain operator ideals

We prove that for some classes of weakly compact operators on the space C[0, 1] the conditions that they are compact, absolutely summing and nuclear are distinct and connected to vector-valued sequence spaces. Received: 22 February 2006     

A Remark on Two Duality Relations

We remark that an easy combination of two known results yields a positive answer, up to log(n) terms, to a duality conjecture that goes back to Pietsch. In particular, we show that for any two symmetric convex bodies K, T in , denoting by N(K, T) the minimal number of translates of T needed to cover K, one has:

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     

Asymptotic Behaviour of Iterates of Volterra Operators on L p (0, 1)

Given k ∈ L¹ (0,1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator V_k defined on L^p (0,1) by

A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

Regularities in Noncommutative Banach Algebras

In this paper we introduce regularities and subspectra in a unital noncommutative Banach algebra and prove that there is a correspondence between them similar to the commutative case. This correspondence involves a radical on a class of Banach algebras equipped with a subspectrum. Taylor and Slodkowski spectra for noncommutative tuples of bounded linear operators are the main examples of subspectra in the noncommutative case.      

Comparability Preserving Maps on Bounded Observables

Let H be a Hilbert space and B _s (H) the set of all self-adjoint bounded linear operators on H. We describe the general form of bijective maps which preserve comparability in both directions. This work was partially supported by a grant from the Ministry of Science of Slovenia.     

The Poisson Kernel for Hardy Algebras

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras H^∞(E), which we call Hardy algebras, and which are noncommutative generalizations of classical H^∞, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which “reproduces” the values, on , of the “functions” coming from H^∞(E). We present results that are natural generalizations of the Poisson integral formula. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the “model space” for the canonical model that can be attached to a point in the disc . We also connect our Poisson kernel to various “point evaluations” and to the idea of curvature. The first named author was supported in part by grants from the National Science Foundation and from the U.S.-Israel Binational Science Foundation. The second named author was supported in part by the U.S.-Israel Binational Science Foundation and by the B. and G. Greenberg Research Fund (Ottawa).     

Some properties and applications of weakly equicompact sets

Let X and Y be Banach spaces. A set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n) in X, there exists a subsequence (x _k(n)) so that (Tx_k(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ↩̸ ℓ₁, of spaces X such that B _X* is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Received: 5 July 2006     

On the Cowen-Douglas Class for Banach Space Operators

In [3], M. J. Cowen and R. G. Douglas prove that the adjoint of a Hilbert space operator T is in the class if and only if T is unitarily equivalent with the operator M _z on a Hilbert space -valued analytic functions, where M _z denotes the operator of multiplication by the independent variable. The proof involves holomorphic vector bundles and Grauert’s theorem. In this paper we use a theorem by I. Gohberg and L. Rodman [4] to give a more elementary proof of this fact, which also works for Banach space operators.      

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.     

Compact AC(σ) Operators

All compact AC(σ) operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact AC(σ) operator into real and imaginary parts.      

Composition Operators on the Space of Bounded Harmonic Functions

We study composition operators on the space of bounded harmonic functions on the open unit disk. The principal goal of this paper is to provide criteria for determining the essential norm of difference of two composition operators. The third author is partially supported by Grant-in-Aid for Scientific Research (No.17540169), Japan Society for the Promotion of Science.     

On Vector Integral Inequalities

The first author introduced an integration theory of vector functions with respect to an operator-valued measure in complete bornological locally convex vector spaces. In this paper some important results behind this Dobrakov-type integration technique in non-metrizable spaces are given. Received: December 10, 2007., Accepted: May 6, 2008.     

On Uniform Exponential Stability of Exponentially Bounded Evolution Families

A result of Barbashin ([1], [15]) states that an exponentially bounded evolution family defined on a Banach space and satisfying some measurability conditions is uniformly exponentially stable if and only if for some 1 ≤ p < ∞, we have that:

Spaces of compact operators and their dual spaces

Theω′-topology on the spaceL(X, Y) of bounded linear operators from the Banach spaceX into the Banach spaceY is discussed in [10]. Let ℒ^w' (X, Y) denote the space of allT∈L(X, Y) for which there exists a sequence of compact linear operators (T _n)⊂K(X, Y) such thatT=ω′−lim_nT_n and let . We show that is a Banach ideal of operators and that the continuous dual spaceK(X, Y)* is complemented in . This results in necessary and sufficient conditions forK(X, Y) to be reflexive, whereby the spacesX andY need not satisfy the approximation property. Similar results follow whenX andY are locally convex spaces. Financial support from the Potchefstroom University and Maseno University is greatly acknowledged. Financial support from the NRF and Potchefstroom University is greatly acknowledged.