Mathematics – Functional Analysis
Scientific paper
2012-03-13
Mathematics
Functional Analysis
22 pages
Scientific paper
In this paper, we introduce $(k,\theta)$-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 $\theta>1$, or $k=1$ and $\theta>e$. 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 $(k,\theta)$-log Bloch space, into the $\mu$-Bloch space ${\mathcal B}^\mu({\Bbb D})$ in terms of the sequence of quotients of the $(k,\theta)$-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 $\mu:{\Bbb D}\rightarrow (0,\infty)$ 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.
Castillo René E.
Clahane Dana D.
Farías-López Juan F.
Ramos-Fernández Julio C.
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