Hankel Operators on Bergman Spaces of Tube Domains over Symmetric Cones

Abstract

We present here some criteria for Schatten-Von Neumann class membership for the small Hankel operator on Bergman space A 2 (TΩ), when TΩ is the tube over the symmetric cone Ω. Mathematics Subject Classification (2000). Primary 47B35; Secondary 32A37, 46E22, 47B10.

On Bergman type integral operators in analytic spaces in tubular domains over symmetric cones

“…As a corollary of one dimensional version of second estimate and first estimate (see, for example, [20], [19,Theorem 3.9]) we obtain the following vital Forelli-Rudin type estimate for Bergman kernel (1.2) which we will use in proof of all our main results [19].…”

“…Replacing above simplyà byL we will get as usual the corresponding larger space of all measurable functions in tube over symmetric cone with the same quazinorm (see [13,19,20]). It is known theà p,q ν (T Ω ) space is nontrivial if and only if ν > n r − 1 (see [5]).…”

“…Various interesting results were provided related with Bergman type integral operators on such type spaces. We refer to [5] and [20] and various references there only for a short portion of such type results.…”

Schatten class Toeplitz operators on weighted Bergman spaces of tube domains over symmetric cones

“…We start with the Hilbert-Schmidt class S 2 := S 2 (A 2 ν (D), H). Proof This result was proved in [18]. As the definition of Bergman spaces here is quite different, let us give a proof here for completeness.…”

“…The first one is the following which is obtained as in [18,Lemma 3.2]. Recalling that ||T || p S p = Tr(T p ) , the proof follows from the fact that for any unit vector(see [20]) g ∈ L 2 (D) , we have and…”