Generalized Berezin transform and Schatten class operators on weighted Bergman spaces [Articol]

Abstract

In this paper we derive certain elementary properties of the generalized Berezin transform Bα, α > −1 and show that if T is a bounded linear operator from the weighted Bergman space L2 a(dAα) into itself, and G(w) =TK(α) wp , w ∈ D, 0 < p < ∞, then Bn 0 G is subharmonic for all n ∈ N and Bn 0 ϕ → ψ, the least harmonic majorant of G. Further, we describe the Schatten class characterization of a bounded linear operator T in terms of the generalized Berezin transform of T. We have also shown that if T ≥ 0 or T is in trace class S1, then tr(T) = (α + 1) Z D (BαT)(z)dλ(z) where dλ(z) = dA(z) (1−|z|2)2 . Further, if A is compact and positive and 0 < p ≤ 1, then we show that Z D Bα(Ap)(z)dλ(z) is finite if and only if ∞X n=1D Aξ(α) n , ξ(α) n E p is finite for some orthonormal basis n ξ(α) n o∞ n=1 in L2 a(dAα).