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Composition operators from $(k, heta)$-logarithmic Bloch spaces to weighted Bloch spaces | | René E. Castillo
; Dana D. Clahane
; Juan F. Farías-López
; Julio C. Ramos-Fernández
; | | Date: |
13 Mar 2012 | | Abstract: | In this paper, we introduce $(k, heta)$-logarithmic Bloch spaces, which
generalize the usual logarithmic Bloch space of the unit ball ${Bbb B}_n$ in
${Bbb C}^n$, where either $k=2$ and $ heta>1$, or $k=1$ and $ heta>e$. After
obtaining norm equivalences among these spaces, we characterize the analytic
self-maps $phi$ of the unit disk ${Bbb D}={Bbb B}_1$ in ${Bbb C}$ that
induce continuous composition operators $C_phi$ from the log-Bloch space
$mathcal{B}^{log}({Bbb D})$, and more generally, a $(k, heta)$-log Bloch
space, into the $mu$-Bloch space ${mathcal B}^mu({Bbb D})$ in terms of the
sequence of quotients of the $(k, heta)$-log Bloch norm of the $n$th power of
$phi$ and the $mu$-Bloch norm of the $n$th power $F_n$ of the identity
function on ${Bbb D}$, where $mu:{Bbb D}
ightarrow (0,infty)$ is
continuous and bounded. We also obtain an expression that is equivalent to the
essential norm of $C_phi$ between these spaces. Our boundedness and essential
norm results together imply characterizations of the compact composition
operators between these spaces. | | Source: | arXiv, 1203.2693 | | Services: | Forum | Review | PDF | Favorites | |
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