A new characterization of Bergman–Schatten spaces and a duality result

Abstract

Let B 0 (D, 2 ) denote the space of all upper triangular matrices A such that lim r→1 − (1 − r 2 ) (A * C (r)) B( 2 ) = 0. We also denote by B 0,c (D, 2 ) the closed Banach subspace of B 0 (D, 2 ) consisting of all upper triangular matrices whose diagonals are compact operators. In this paper we give a duality result between B 0,c (D, 2 ) and the Bergman-Schatten spaces L 1 a (D, 2 ). We also give a characterization of the more general Bergman-Schatten spaces L p a (D, 2 ), 1 p < ∞, in terms of Taylor coefficients, which is similar to that of M. Mateljevic and M. Pavlovic [M. Mateljevic, M. Pavlovic, L p -behaviour of the integral means of analytic functions, Studia Math. 77 (1984) 219-237] for classical Bergman spaces.

Some new characterizations of Bloch type spaces of infinite matrices via Schur multipliers

“…Let 0 ≤ r < 1. In [14] (see also [18], Theorem 7.11) it is proved that B(D, ℓ 2 ) = L 1 a (D, ℓ 2 ) * by using the duality pair…”

“…We recall the matrix versions of Bloch and Bergman spaces (see e.g. [19], [14] and [18]). For a treatment of a more general case of Bloch and Bergman spaces in the case of vector valued functions we refer to [3], [6] and [7].…”