Hankel Operators on Weighted Bergman Spaces

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. Introduction. This paper deals with the connections between an analytic function f on the unit disc A in the complex plane and the membership in some Schatten ideal of the Hankel operator that f induces on certain weighted Bergman spaces of analytic functions on A. The general theme is the interplay between smoothness or growth conditions on f and the size of the s-numbers of the Hankel operator induced by f. Such themes have appeared before in connection with Hankel operators on the Hardy space H2 and a portion of the results here parallel those. These results may be found in the papers of V. V. Peller [24], [25], [26], R. Rochberg [27], [28], and S. Semmes [31]. In our context, however, an entirely new phenomena occurs at the lower end point, where the Macaev ideal appears in place of the nuclear operators. The appearance of the Macaev ideal requires the introduction of some new interpolation results to prove the intermediate results since the lower end point condition is not properly structured for interpolation. Important roles throughout the paper are played by the Mobius group and various spaces of functions invariant under it.The paper is arranged in the following manner. Section 1 introduces all the relevent definitions and background material and closes with statement of the main theorem. Section 2 is devoted to general properties of Hankel operators and describes an invariance of the action of the Mobius group on a Hankel operator. Section 3 contains the description of the Hankel operators with monomial symbols acting on other monomials and, as a consequence, establishes an isometry between the Dirichlet space and those Hankel operators that are Hilbert-Schmidt. Section 4 gives the characterization of bounded and compact Hankel operators; this result was nroved first by S. Axler [6] in a special but representative case. Section 5

Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids

“…Les propriétés spectrales des opérateurs de Hankel sont traités dans [2] et [3] pour les espaces de Bergman et dans l'ouvrage de [14] pour l'espace de Hardy. On pourra aussi consulter le livre de Zhu [19] et les références qu'il contient.…”

Hankel Operators on Standard Bergman Spaces

“…It has been a lot of activity in the theory of Hankel operators on Bergman spaces in recent years, and this topic has become a classical theme in complex analysis and operator theory (see for example [1], [3], [4], [10], [11], [13], and [17]). For Hankel operators with conjugate analytic symbols, that is Hf with f ∈ A 2 α , one has that Hf is bounded on A 2 α if and only if the symbol f belongs to the Bloch space; Hf is compact if and only if f belongs to the little Bloch space (see [1], [2]); and the membership in Schatten p-classes of the Hankel operator Hf is equivalent to the function f being in the analytic Besov space B p for 1 < p < ∞, and to f being constant when 0 < p ≤ 1.…”

“…For Hankel operators with conjugate analytic symbols, that is Hf with f ∈ A 2 α , one has that Hf is bounded on A 2 α if and only if the symbol f belongs to the Bloch space; Hf is compact if and only if f belongs to the little Bloch space (see [1], [2]); and the membership in Schatten p-classes of the Hankel operator Hf is equivalent to the function f being in the analytic Besov space B p for 1 < p < ∞, and to f being constant when 0 < p ≤ 1. Therefore, for conjugate analytic symbols, the picture on the boundedness, compactness and Schatten p-classes is complete.…”

“…Therefore, for conjugate analytic symbols, the picture on the boundedness, compactness and Schatten p-classes is complete. However, the proof of the necessity and sufficiency of the condition f ∈ B p for Hf being in the Schatten class S p when 1 < p < 2 given in [1] is rather difficult and technical, and it is our aim to provide a more "elementary" proof of that result. Theorem 1.…”

Index of Γ-Equivariant Toeplitz Operators

“…But in this case the principal function for a pair of Toeplitz operators that commute modulo the trace ideal in B(H t ) is well understood -it is basically given by the fact that the index of T z is equal to one ( [1]) and gives explicit formulas that lead to the results stated above.…”