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Boundedness and Compactness of Generalized Hankel Operators on Bounded Symmetric Domains
 

Summary: Boundedness and Compactness of Generalized Hankel
Operators on Bounded Symmetric Domains
Jonathan Arazy
(Journal of Functional Analysis, Vol. 137 (1996), 97-151)
Abstract
The map b ! H b := (I P )M b P from analytic functions on the unit disk D to the
associated Hankel operators on the Hardy space or the weighted Bergman spaces is known to
be an important tool in studying the \size" of the function b in terms of the \size" of H b .
Moreover, this map is equivariant, namely it intertwines the natural actions of the Mobius
group Aut(D) on functions and operators. This theory extends to some extent to the context
of the open unit ball B n in C n , but it fails in Cartan domains of rank r > 1, because in this
case the map b ! H b trivializes as H b is compact only if H b = 0 (and b is constant).
We study generalizations A b of Hankel operators in the context of weighted Bergman spaces
over a Cartan domain of tube type with rank r > 1. The map b ! A b is equivariant and non-
trivial. We study also in this context generalized Bloch and little Bloch spaces (B and B 0
respectively), and generalized BMOA and VMOA spaces with respect to weighted volume
measure.
The main results are that A b is bounded if and only if b 2 B if and only if b 2 BMOA,
and A b is compact if and only if b 2 B 0 if and only if b 2 VMOA.
1 Introduction

  

Source: Arazy, Jonathan - Department of Mathematics, University of Haifa

 

Collections: Mathematics