For p > 0, let B p (B n ) and L p (B n ) respectively denote the p-Bloch and holomorphic p-Lipschitz spaces of the open unit ball B n in C n . It is known that B p (B n ) and L 1−p (B n ) are equal as sets when p ∈ (0, 1). We prove that these spaces are additionally norm-equivalent, thus extending known results for n = 1 and the polydisk. As an application, we generalize work by Madigan on the disk by investigating boundedness of the composition operator C φ from L p (B n ) to L q (B n ).

Let H 2 (D) denote the Hardy space of a bounded symmetric domain D ⊂ C n in its standard Harish-Chandra realization, and let A p α (D) be the weighted Bergman space with p ≥ 1 and α < ε D , where ε D is a critical value depending on D., then φ has a unique fixed point z 0 in D. We then prove that the spectrum of C φ as an operator on these function spaces is precisely the set consisting of 0, 1, and all possible products of eigenvalues of φ (z 0 ). These results extend previous work by Caughran/Schwartz and MacCluer. As a corollary, we now have that MacCluer's previous spectrum results on the unit ball B n extend to H p (∆ n ) (not only for p = 2 but for all p > 1) and A p α (∆ n ) (for p ≥ 1), where ∆ n is the polydisk in C n .

a b s t r a c tWe characterize the analytic self-maps / of the unit disk D in C that induce continuous composition operators C / from the log-Bloch space B log ðDÞ to l-Bloch spaces B l ðDÞ in terms of the sequence of quotients of the l-Bloch semi-norm of the nth power of / and the log-Bloch semi-norm of the nth power F n of the identity function on D, where l : D ! ð0; 1Þ is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of C / between these spaces, thus characterizing / such that C / 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 / acting on B log to the case when C / acts on these more general log-Bloch-type spaces.Published by Elsevier Inc.