A remark on the isomorphic classification of
weighted spaces of homolorphic functions on the upper half plane

Abstract. We complete the isomorphic classification for weighted spaces \(H_v( \mathbb G)\) of holomorphic functions on the upper half plane \( \mathbb G\) with respect to standard weights \(v\) which are of at most moderate growth. We show that there are only two isomorphism classes for the corresponding Banach spaces \(H_v( \mathbb G)\), namely \( l_{ \infty}\) and \(H_{ \infty}\). We prove that \(H_v( \mathbb G)\) is isomorphic to \( H_{ \infty}\) if and only if \(v\) grows slowly. In particular \(H_v( \mathbb G)\) is isomorphic to \(H_{ \infty}\) if \(v\) is bounded.

Key words and phrases. Weighted Banach spaces, holomorphic functions, upper half plane.