TOEPLITZ AND HANKEL OPERATORS ON WEIGHTED BERGMAN SPACES

Abstract

In this paper we have shown that if S ∈ L(L 2 a (dAα)) and Θ (α) S (x, y)Θ(α) T (x, y)(K(α) (x, y))2 ≈ Θ (α) ST (x, y)(K(α) (x, y))2 for all x, y ∈ D and for all T ∈ L(L 2 a (dAα)), then S = T (α) φ for some φ ∈ H∞(D) and the matrix of S is lower triangular, where Θ(α) S (x, y) for S ∈ L(L 2 a (dAα)) is a function on D × D meromorphic in x and conjugate meromorphic in y. Further, we show that if ψ, φ ∈ L∞(D), R(α) ∈ L(L 2 a (dAα)), then Θ(α) T (α) φ (x, y)Θ(α) S (α) ψ (x, y)(K(α) (x, y))2 ≈ Θ (α) R(α) (x, y) ·(K(α) (x, y))2 holds for all x, y ∈ D if and only if there exists β ∈ C such that φ ≡ β and R(α) = S (α) βψ .