We characterize the analytic self-maps @f of the unit disk D in C that induce continuous composition operators C"@f from the log-Bloch space B^l^o^g(D) to @m-Bloch spaces B^@m(D) in terms of the sequence of quotients of the @m-Bloch semi-norm of the nth power of @f and the log-Bloch semi-norm of the nth power F"n of the identity function on D, where @m:D->(0,~) is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of C"@f between these spaces, thus characterizing @f such that C"@f is compact. After finding a pairwise norm equivalent family of log-Bloch type spaces that are defined on the unit ball B"n of C^n and include the log-Bloch space, we obtain an extension of our boundedness/compactness/essential norm results for C"@f acting on B^l^o^g to the case when C"@f acts on these more general log-Bloch-type spaces.
['René Erlín Castillo', 'Dana D. Clahane', 'Juan F. Farías López', 'Julio César Ramos Fernández']
Composition operators from logarithmic Bloch spaces to weighted Bloch spaces