Boundedness from Below of Composition Operators on α-Bloch Spaces

Abstract

Abstract. We give a necessary and sufficient condition for a composition operator on an α-Bloch space with α ≥ 1 to be bounded below. This extends a known result for the Bloch space due to P. Ghatage,

Composition operators on µ-Bloch type spaces

“…In this case, the composition operator C ϕ is bounded for all symbols ϕ. The same is true if μ(t) = t α with α > 1 (see [3]). The case μ(t) = t log 2 t , is considered in [17,Theorem 1].…”

“…On the other hand, Gathage, Zheng and Zorboska [8] characterized closed range composition operators on the Bloch space. This result has been extended by Chen and Gauthier [3] to α-Bloch spaces. Composition operators between α-Bloch and/or Lipschitz spaces on the unit ball are studied in [5], while the case of the polydisk was thoroughly studied in [6].…”

“…Now, we present a necessary and sufficient condition for a composition operator on B μ to be bounded below (and therefore with closed range). The purpose here is to generalize the results in [8,3] to the μ-Bloch space. To this end, for ε > 0, let us denote…”

“…The purpose here is to obtain genuine extensions of the results in [17,8,3]. For that reason, throughout this section we will assume that μ : (0, 1] → R is a positive continuous function satisfying μ(t) → 0 as t → 0 + , and also that μ can be extended to a complex function μ satisfying the following properties:…”

“…We will approach these problems from a slightly different point of view. In the first place, we will consider only those functions μ that can be extended to non vanishing, complex valued holomorphic functions, that satisfy a reasonable, for the purpose of extending the results of [17,8,3], geometric condition on the Euclidean disk D(1, 1). In Section 3, we will consider the problem of compactness of composition operators acting on μ-Bloch spaces.…”