An Integration by Parts Formula in Sobolev Spaces

We prove the following formula

Comparison of two kinds of Hankel type operators

We compare two kinds of Hankel type operators, the “BJP-operators” introduced by Boman—Janson—Peetre and the “Z-operators” considered by Zhang. We show that every BJP-operator comes as an orthogonal sum ofZ-operators.     

Toeplitz Operators and Herz Spaces on the Half-Space

On the setting of the half-space we introduce the Schatten-Herz classes of Toeplitz operators and obtain characterizations for positive Toeplitz operators to belong to those classes. We also prove results concerning the boundedness and compactness of Toeplitz operators with Herz symbols. Such a study has been recently done on the ball. At a critical step of the proofs we employ a much simplified argument to extend the range of parameters for Herz spaces on which the Berezin transform is bounded. Our results show not only that most of results on the ball continue to hold, but also that there is some pathology caused by the unboundedness of the domain. The first author was in part supported by a Korea University Grant(2007), the second author was in part supported by Hanshin University Research Grant, and both authors were in part supported by KOSEF(R01-2003-000-10243-0).     

Unconditionality in spaces of smooth functions

Our main result shows that subspaces of L¹([0, 1]) on which the blow-up operators act compactly are isometric to dual spaces, and their natural predual belongs to the Banach-Mazur closure of quotient spaces of . Related general results are shown for subspaces X of or of reflexive K?the function spaces, which imply that when X consists of smooth functions it embeds into a Banach space with an unconditional basis. Received: 25 September 2008     

A class of linear functionals on the Bergman space over the complement of the unit lattice

We prove that the norm of the functional where f is a rational function, is equal to 1. Supported by MNZŽS Serbia, Project No. 144010. Received: 21 August 2006     

On very weak entire solutions of nonlinear partial differential equations

Let us consider the variational equation in R ⁿ

Approximate Factorization in Generalized Hardy Spaces

In this paper we establish a connection between the approximate factorization property appearing in the theory of dual algebras and the spectral inclusion property for a class of Toeplitz operators on Hilbert spaces of vector valued square integrable functions. As an application, it follows that a wide range of dual algebras of subnormal Toeplitz operators on various Hardy spaces associated to function algebras have property (A ₁(1)). It is also proved that the dual algebra generated by a spherical isometry (with a possibly infinite number of components) has the same property. One particular application is given to the existence of unimodular functions sitting in cyclic invariant subspaces of weak* Dirichlet algebras. Moreover, by this method we provide a unified approach to several Toeplitz spectral inclusion theorems. Research partially supported by grant CNCSIS GR202/2006 (cod 813).     

On the action of Lipschitz functions on vector-valued random sums

Let X be a Banach space and let (ξ_j)_j ≧ 1 be an i.i.d. sequence of symmetric random variables with finite moments of all orders. We prove that the following assertions are equivalent:

Products of Hankel operators

This paper studies products of Hankel operators on the Hardy space. We show thatH _f ^(1)* H _f(2) H _f ^(3)* =0 for all permutation if and only if eitherH _f1 orH _f2 orH _f3 is zero. Using Douglas' localization theorem and Izuchi's theorem on Sarason's three functions problem, we show that

Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions

Multiplication operators in weighted Banach (and locally convex) spaces of functions holomorphic in the unit disc are well known. In this note we investigate the connection between power boundedness, mean ergodicity and uniform mean ergodicity of such operators. Received: 13 October 2008, Revised: 18 November 2008     

Properties of generalized Toeplitz operators

An operatorX: is said to be a generalized Toeplitz operator with respect to given contractionsT ₁ andT ₂ ifX=T ₂XT₁ ^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.Dedicated to our friend Cora Sadosky on the occasion of her sixtieth birthday     

Unitary invariants of spectral measures with the CGS-property

A «CGS-property» for the spectral measures is introduced and the classical results of determining complete systems of unitary invariants for self-adjoint and bounded normal operators on separable Hilbert spaces are extended to the class of spectral measures with this property. As a consequence, the above mentioned results are extended to unbounded normal operators on separable Hilbert spaces. Moreover, three different kinds of multiplicity are defined and it is shown that for the measures with the «CGS-property» they all coincide. In the last section some analogues of the multiplicity functions defined by Stone [14] are related to the total multiplicity.     

Toeplitz operators and arguments of analytic functions

For a Toeplitz operator T _φ , we study the interrelationship between smoothness properties of the symbol φ and those of the functions annihilated by T _φ . For instance, it follows from our results that if φ is a unimodular function on the circle lying in some Lipschitz or Zygmund space Λ^α with 0 < α < ∞, and if f is an H ^p -function (p ≥ 1) with T _φ f = 0, then f ∈ Λ^α and

Common fixed points in cone metric spaces

In this paper we consider a notion of g-weak contractive mappings in the setting of cone metric spaces and we give results of common fixed points. This results generalize some common fixed points results in metric spaces and some of the results of Huang and Zhang in cone metric spaces. Supported by Universitá degli Studi di Palermo, R. S. ex 60%.     

Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces

We prove that Lipschitz mappings are dense in the Newtonian–Sobolev classes N ^1,p (X, Y) of mappings from spaces X supporting p-Poincaré inequalities into a finite Lipschitz polyhedron Y if and only if Y is [p]-connected, π ₁(Y) = π ₂(Y) = · · · = π _[p](Y) = 0, where [p] is the largest integer less than or equal to p. This work was supported by the NSF grant DMS-0500966.     

Compact Differences of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article, we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.      

Sur un example de categorie presque-abelienne

Résumé  Si Λ est la catégorie des espaces localement convexes séparés et complets, nous montrons que la catégorie duale de Λ est une catégorie presque-abélienne et par suite, la catégorie Λ l’est aussi.

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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.