Composition operators from logarithmic Bloch spaces to weighted Bloch spaces

Abstract

We characterize the analytic self-maps ϕ of the unit disk D in C that induce continuous composition operators Cϕ from the log-Bloch space Blog(D) to μ-Bloch spaces Bμ(D) in terms of the sequence of quotients of the μ-Bloch semi-norm of the nth power of ϕ and the log-Bloch semi-norm of the nth power Fn of the identity function on D, where μ:D→(0,∞) is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of Cϕ between these spaces, thus characterizing ϕ such that Cϕ is compact. After finding a pairwise norm equivalent family of log-Bloch type spaces that are defined on the unit ball Bn of Cn and include the log-Bloch space, we obtain an extension of our boundedness/compactness/essential norm results for Cϕ acting on Blog to the case when Cϕ acts on these more general log-Bloch-type spaces.