Linear Isometries of some Normed Spaces of Analytic Functions

Extract

For 1 ≦ p < ∞ let H^p denote the familiar Hardy space of analytic functions on the open unit disc D and let ‖·‖ denote the H^p norm. Let S^p denote the space of analytic functions f on D such that f′ ∊ H^p. In this paper we will describe the linear isometries of S^p into itself when S^p is equipped with either of two norms. The first norm we consider is given by

(1)

and the second by

(2)

(It is well known [1, Theorem 3.11] that f′ ∊ H^p implies continuity for f on D, the closure of D. Thus (2) actually defines a norm on S^p.) In the former case, with the norm defined by (1), we will show that an isometry of S^p induces, in a sense to be made precise in Section 2, an isometry of H^p and that Forelli's characterization [2] of the isometries of H^p can thus be used to describe the isometries of H^p.