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

Let B 0 ( D , ℓ 2 ) denote the space of all upper triangular matrices A such that lim r → 1 − ( 1 − r 2 ) ‖ ( A ∗ C ( r ) ) ′ ‖ B ( ℓ 2 ) = 0 . We also denote by B 0 , c ( D , ℓ 2 ) the closed Banach subspace of B 0 ( D , ℓ 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 , ℓ 2 ) and the Bergman–Schatten spaces L a 1 ( D , ℓ 2 ) . We also give a characterization of the more general Bergman–Schatten spaces L a p ( D , ℓ 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.