2012/03/13 by Castillo, René E., Clahane, Dana D., Farías-López, Juan F. +1
#30H30 #32A18 #32A30 #47A55 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 47B33 #Secondary: 30D45
paper · doi:10.48550/arxiv.1203.2693
We characterize the analytic self-maps ϕ of the unit disk \Bbb D in \Bbb C that induce continuous composition operators Cϕ from the log-Bloch space Blog(\Bbb D) to μ-Bloch spaces \mathcal Bμ(\Bbb 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 (norm) of the nth power Fn of the identity function on \Bbb D, where μ:\Bbb 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 \Bbb Bn of \Bbb Cn and include the log-Bloch space, we obtain an extension of our boundedness/compactness/essential norm results for Cϕ acting on \mathcal Blog to the case when Cϕ acts on these more general log-Bloch-type spaces.