The commutant of rationally cyclic subnormal operators and rational approximation

Let denote the set of analytic bounded point evaluations forR ^q (K, ). Assume that . In this paper, we first show that if is a finitely connected domain and if the evaluation map fromR ^q (K, )L () toH () is surjective, then | is absolutely continuous with respect to harmonic measure for . This generalizes Olin and Yang's corresponding result for polynomials and the proof we present here is simpler. We also provide an example that shows this absolute continuity property fails in general when is an infinitely connected domain. In the second part, we then offer a solution to a problem of Conway and Elias.     

A Note on the Range of Toeplitz Operators

A well known lemma attributed to Coburn states that a Toeplitz operator with non-trivial kernel acting on the Hardy space must have dense range. We show that the range of a non-zero Toeplitz operator with non-trivial kernel must contain all polynomials and state this in a precise form.     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

Composition Operators on Small Spaces

We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro’s theorem concerning compact composition operators on small spaces. We also prove the converse of Shapiro’s theorem if the symbol function is already contained in the space under consideration. In the course of the proofs we characterize the bounded composition operators on the Zygmund class. Also, as a by-product of our arguments, we show that small holomorphic Sobolev spaces are algebras.     

Spectra of Composition Operators on BMOA

It is shown that if ϕ is a univalent self-map on the unit disk is not an automorphism and has a fixed point in and if the essential spectral radius of the composition operator C_ϕ on H² is different from zero, then the spectrum of C_ϕ on BMOA coincides with This answers in the affirmative a conjecture by MacCluer and Saxe.     

Herz Classes and Toeplitz Operators in the Disk

In this paper we decompose into diadic annuli and consider the class S_p,q of Toeplitz operators T_φ for which the sequence of Schatten norms belongs to ℓ^q, where φ_n = φχ A_n. We study the boundedness and compactness of the operators in S_p,q and we describe the operators T_φ , φ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols φ.     

Operators on harmonic Bergman spaces

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL ^p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

An Operator Transform from Class A to the Class of Hyponormal Operators and its Application

In this paper, we shall give an operator transform from class A to the class of hyponormal operators. Then we shall show that and in case T belongs to class A. Next, as an application of we will show that every class A operator has SVEP and property (β).     

A Note on the Spectrum of Invertible p-hyponormal Operators

A bounded linear operator T is clalled p-hyponormal if (T*T)^p ≥ (TT)^p, 0 < p < 1. It is known that for semi-hyponormal operators (p = 1/2), the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. In this paper we prove a somewhat weaker result for invertible p-hyponormal operators for 0 < p < 1/2.     

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

Composition Operators on Orlicz–Lorentz Spaces

In this paper, we study the boundedness and the compactness of composition operators on Orlicz–Lorentz spaces.      

On the Range of Elementary Operators

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A₁,A₂,.., A_n) and B = (B₁, B₂,.., B_n) be n-tuples in B(H), we define the elementary operator by In this paper we initiate the study of some properties of the range of such operators.     

Compact Operators on Bergman Spaces

We prove that a bounded operator S on L _a ^p for p > 1 is compact if and only if the Berezin transform of S vanishes on the boundary of the unit disk if S satisfies some integrable conditions. Some estimates about the norm and essential norm of Toeplitz operators with symbols in BT are obtained.     

Nearly Invariant Subspaces Related to Multiplication Operators in Hilbert Spaces of Analytic Functions

We consider Hilbert spaces of analytic functions defined on an open subset of , stable under the operator M_u of multiplication by some function u. Given a subspace of which is nearly invariant under division by u, we provide a factorization linking each element of to elements of on the inverse image under u of a certain complex disc, for which we give a relatively simple formula. By applying these results to and u(z) = z, we obtain interesting results involving a H²-norm control. In particular, we deduce a factorization for the kernel of Toeplitz operators on Dirichlet spaces. Finally, we give a localization for the problem of extraneous zeros.Submitted: January 18, 2003 Revised: December 20, 2003     

Spectral measures,III: Densely defined spectral measures

In this paper we extend the theory of spectral measures developed in Parts I and II to the case where values are assumed in the set of discontinuous (in normed spaces „unbounded”) operators. Examples of operators in nonlocally convex spaces are given, which have densely defined measures.     

Bounded and compact multipliers between Bergman and Hardy spaces

This paper studies the boundedness and compactness of the coefficient multiplier operators between various Bergman spacesA ^p and Hardy spacesH ^q . Some new characterizations of the multipliers between the spaces with exponents 1 or 2 are derived which, in particular, imply a Bergman space analogue of the Paley-Rudin Theorem on sparse sequences. Hardy and Bergman spaces are shown to be linked using mixed-norm spaces, and this linkage is used to improve a known result on (A ^p ,A ²), 1<p<2.Compact (H ¹,H ²) and (A ¹,A ²) multipliers are characterized. The essential norms and spectra of some multiplier operators are computed. It is shown that forp>1 there exist bounded non-compact multiplier operators fromA ^p toA ^q if and only ifpq.     

Norms of Toeplitz and Hankel Operators on Hardy Type Subspaces of Rearrangement-Invariant Spaces

We prove analogues of the Brown-Halmos and Nehari theorems on the norms of Toeplitz and Hankel operators, respectively, acting on subspaces of Hardy type of reflexive rearrangement-invariant spaces with nontrivial Boyd indices.     

Extremal Problems for Operators in Banach Spaces Arising in the Study of Linear Operator Pencils

Inspired by some problems on fractional linear transformations the authors introduce and study the class of operators satisfying the condition where stands for the spectral radius; and the class of Banach spaces in which all operators satisfy this condition, the authors call such spaces V-spaces. It is shown that many well-known reflexive spaces, in particular, such spaces as L_p(0,1) and C_p, are non-V-spaces if p 2; and that the spaces l_p are V-spaces if and only if 1 < p < . The authors pose and discuss some related open problems.     

On the Spectrum of Invertible Semi-hyponormal Operators

It is known that for a semi-hyponormal operator, the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. The original proof of this theorem involves the so-called singular integral model. The purpose of this paper is to give a different proof of the same theorem for the case of invertible semi-hyponormal operators without using the singular integral model.      

Singular Integral Operators with Carleman Shift and Discontinuous Coefficients in the Spaces H^{\omega}_0(\Gamma,\rho)

For the singular integral operators with Carleman shift, preserving or changing orientation, and piecewise continuous coefficients we prove the theorem on Fredholmness and obtain the formula for index in the generalized Hölder spaces defined by an arbitrary continuity modulus from the Bari-Stechkin class and some general weights on a closed or open finite Lyapunov curve .