Boundedness and Compactness of Generalized Hankel Operators on Bounded Symmetric Domains

Abstract

The mapb→Hb:=(I−P)MbPfrom analytic functions on the unit diskDto 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 functionbin terms of the “size” of Hb. Moreover, this map is equivariant, namely it intertwines the natural actions of the Möbius groupAut(D) on functions and operators. This theory extends to some extent to the context of the open unit ballBninCn, but it fails in Cartan domains of rankr>1, because in this case the mapb→Hbtrivializes as Hbis compact only if Hb=0 (andbis constant). We study generalizations Abof Hankel operators in the context of weighted Bergman spaces over a Cartan domain of tube type with rankr>1. The mapb→Abis equivariant and non-trivial. We study also in this context generalized Bloch and little Bloch spaces (B and B0respectively), and generalized BMOA and VMOA spaces with respect to weighted volume measure. The main results are that Abis bounded if and only ifb∈B if and only ifb∈BMOA, and Abis compact if and only ifb∈B0if and only ifb∈VMOA.