Compact Hankel Forms on Planar Domains

A Hankel form on a Hilbert function space is a bounded, symmetric, bilinear form [., .] satisfying [fx, y] = [x, fy] for a class of multipliers f. We prove analogs of Weyl–Horn and Ky Fan inequalities for compact Hankel forms, and apply them to estimate the related eigenvalues, both for Hardy–Smirnov and Bergman spaces norms associated to multiply connected planar domains. In the case of the unit disk, we investigate the asymptotic of some measures constructed by eigenfunctions of Hankel operators with certain Markov functions as symbols. Submitted: May 2, 2008. Accepted: June 28, 2008.     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

Approximately Local Derivations

Certain linear operators from a Banach algebra A into a BanachA-bimodule X, which are called approximately local derivations,are studied. It is shown that when A is a C^*-algebra, a Banachalgebra generated by idempotents, a semisimple annihilator Banachalgebra, or the group algebra of a SIN or a totally disconnectedgroup, bounded approximately local derivations from A into Xare derivations. This, in particular, extends a result of B.E. Johnson that ‘local derivations on C^*-algebras arederivations’ and provides an alternative proof of it.     

The Grunsky Norm and Some Coefficient Estimates for Bounded Functions

We estimate the growth order in n of the Taylor coefficientsof bounded univalent functions in the unit disk. Our estimatedepends on the norm of the Grunsky operator. In particular,we improve the known results for coefficients of quasiconformallyextendible univalent functions. 1991 Mathematics Subject Classification30C50, 30C75, 30C55.     

The Amenability of Measure Algebras

In this paper we shall prove that the measure algebra M(G) ofa locally compact group G is amenable as a Banach algebra ifand only if G is discrete and amenable as a group. Our contributionis to resolve a conjecture by proving that M(G) is not amenablein the case where the group G is not discrete. Indeed, we shallprove a much stronger result: the measure algebra of a non-discrete,locally compact group has a non-zero, continuous point derivationat a certain character on the algebra.     

Homological Properties of Fully Bounded Noetherian Rings

Let R be a fully bounded Noetherian ring of finite global dimension.Then we prove that K dim (R) gldim (R). If, in addition, Ris local, in the sense that R/J(R) is simple Artinian, thenwe prove that R is Auslander-regular and satisfies a versionof the Cohen–Macaulay property. As a consequence, we showthat a local fully bounded Noetherian ring of finite globaldimension is isomorphic to a matrix ring over a local domain,and a maximal order in its simple Artinian quotient ring.     

On the Auslander-Reiten Periodicity of Self-Injective Algebras

This paper investigates how to relate the syzygy periodicityof a self-injective algebra A to its Auslander–Reitenperiodicity. Moreover, a characterization is provided of theAuslander–Reiten bounded A–A-bimodules that areperiodic. 2000 Mathematics Subject Classification 16G70, 16E40(primary), 16G20 (secondary).     

Auslander-Regular Algebras and Maximal Orders

Let R be an Auslander-regular, Cohen-Macaulay, Noetherian ringthat is stably free. Then, we prove that R is a domain and amaximal order in its division ring of fractions. In particular,this applies to the Sklyanin algebra S and shows that, whenS satisfies a polynomial identity, it is actually a finite moduleover its centre.     

Gleason Property and Extensions of States on Projection Logics

We prove that every state on the projection logic P(M) of avon Neumann algebra M not containing a direct summand of typeI₂ extends to a state of an arbitrary larger unital logic L.We also show that if a C^*-algebra enjoys the Gleason property,and if it possesses sufficiently many projections, then an analogousresult can be derived. Moreover, we prove that the extensionscan be taken linear in a complete order unit norm space associatedwith L. (Results of this paper generalize results of [22] andmay contribute to the noncommutative measure theory, convextheory of state spaces and foundations of quantum physics.)     

The Berezin transform and Laplace-Beltrami operator

We study the Berezin transform of bounded operators on the Bergman space on a bounded symmetric domain Ω in Cn. The invariance of range of the Berezin transform with respect to G=Aut(Ω), the automorphism group of biholomorphic maps on Ω, is derived based on the general framework on invariant symbolic calculi on symmetric domains established by Arazy and Upmeier. Moreover we show that as a smooth bounded function, the Berezin transform of any bounded operator is also bounded under the action of the algebra of invariant differential operators generated by the Laplace-Beltrami operator on the unit disk and even on the unit ball of higher dimensions.     

多重调和Bergman空间上的Toeplitz算子和Hankel算子

完全刻画多重调和Bergman空间上Toeplitz算子和Hankel算子的紧性.运用紧Toeplitz算子这个结果,建立了Toeplitz代数和小Hankel代数的短正合列,推广了单位圆盘上相应的结果.     

Products of Commuting Boolean Algebras of Projections and Banach Space Geometry

New criteria and Banach spaces are presented (for example, GL-spacesand Banach spaces with property () that ensure that the Booleanalgebra generated by a pair of bounded, commuting Boolean algebrasof projections is itself bounded. The notion of R-boundednessplays a fundamental role. It is shown that the strong operatorclosure of any R-bounded Boolean algebra of projections is necessarilyBade complete. Also, for a Dedekind -complete Banach latticeE, the Boolean algebra consisting of all band projections inE is R-bounded if and only if E has finite cotype. In this situation,every bounded Boolean algebra of projections in E is R-boundedand has a Bade complete strong closure. 2000 Mathematics SubjectClassification 46B20, 47L10 (primary), 46B42, 47B40, 47B60 (secondary).     

Bloch Functions in Several Complex Variables, I

Bloch functions on the unit disk are those analytic functionsf for which the quantity |f'(z)|(1 – |z|²) is bounded(for z in the disk). In this paper, Bloch functions on boundedhomogeneous domains in complex m-space are studied. The mainresult shows that many of the equivalent definitions of Blochfunctions on the unit disk are also equivalent in the generalsetting. Current Address: 39 Trinity College, Dublin 2, Ireland.     

Lacunary series and pseudocontinuations

The main goal of this paper is to prove the following statement. Let be a function holomorphic and of bounded characteristic in the unit disk , where E is a Λ(1)-subset of ℤ₊. Assume that f has a pseudocontinuation of bounded characteristic to the annulus {zɛℂ:1<|z|<R}. Then f admits analytic continuation to the disk R . In particular, f is a polynomial if R=+∞. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 16–32. Translated by A. B. Aleksandrov.     

Truncations of multilinear Hankel operators

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are still bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

     

Zel'Manov's Theorem for Primitive Jordan-Banach Algebras

The following result is well known and easy to prove (see [14,Theorem 2.2.6]). Theorem 0. If A is a primitive associative Banach algebra, thenthere exists a Banach space X such that A can be seen as a subalgebraof the Banach algebra BL(X) of all bounded linear operatorson X in such a way that A acts irreducibly on X and the inclusionABL(X) is continuous. In fact, if X is any vector space on which the primitive Banachalgebra A acts faithfully and irreducibly, then X can be convertedin a Banach space in such a way that the requirements in Theorem0 are satisfied and even the inclusion ABL(X) is contractive. Roughly speaking, the aim of this paper is to prove the appropriateJordan variant of Theorem 0.     

On Connections Between Hankel, Laguerre and Heisenberg Multipliers

We prove two results showing connections between Hankel multipliers,multipliers for a certain kind of Laguerre expansion and multipliersfor the Heisenberg sublaplacian. In their spirit they are closeto the well-known theorems of de Leeuw and Igari relating L^pmultipliers on R and its discrete subgroup Z. Also a conjectureis stated and its consequences are discussed.     

Bounded Solutions of Some Second Order Nonlinear Differential Equations

We prove Landesman–Lazer type existence conditions forthe solutions bounded on the real line, together with theirfirst derivatives, for some second order nonlinear differentialequations of the form x'+cx'+f(t, x)=0. The proofs use upperand lower solutions techniques together with some limiting arguments.     

Chracters on Weighted Amenable Groups

In this paper we prove that for every weight on an amenablegroup there is always a continuous bounded character on thatgroup. Thus we may assume that any weight on an amenable groupis always greater than 1. Using a result of N. Grønbæk[1], this implies that the only amenable weighted group algebrasare up to isomorphism L¹(G)for some amenable group G. A Hahn–Banachtype generalisation is given for the extension of bounded charactersand examples are given showing that the assumption of amenabilityis necessary.     

SEMISIMPLE BANACH ALGEBRAS GENERATED BY STRONGLY CONTINUOUS REPRESENTATIONS OF MOORE GROUPS

We prove that if is a strongly continuous bounded representationof a Moore group G on a Banach space X, and if the Arveson spectrumof is scattered, then the closure with respect to the weakoperator topology in L(X) of the algebra generated by the transforms _Gf(t) (t)d t with f L¹(G) is a semisimple Banach algebra.