The map b → ℋb:= (I - P) Mb̄P 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 ℋb. Moreover, this map is equivariant, namely it intertwines the natural actions of the Möbius group Aut(D) on functions and operators. This theory extends to some extent to the context of the open unit ball Bn in Cn, but it fails in Cartan domains of rank r > 1, because in this case the map b → ℋb trivializes as ℋb is compact only if ℋb = 0 (and b is constant). We study generalizations script capital A signb of Hankel operators in the context of weighted Bergman spaces over a Cartan domain of tube type with rank r > 1. The map b → script capital A signb is equivariant and non-trivial. We study also in this context generalized Bloch and little Bloch spaces (ℬ and ℬ0 respectively), and generalized ℬℳscript capital O signscript capital A sign and script capital V signℳscript capital O signscript capital A sign spaces with respect to weighted volume measure. The main results are that script capital A signb is bounded if and only if b∈ℬ if and only if b∈ℬℳscript capital O signscript capital A sign, and script capital A signb is compact if and only if b∈ℬ0 if and only if b∈script capital V sign ℳscript capital O signscript capital A sign.
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