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Weighted composition operators on the Bloch space. (English) Zbl 0985.47022
Let $D$ be the open disk of the complex plane. Then the Bloch space ${\cal B}$ on $D$ and the little Bloch space ${\cal B}_0$ on $D$ are defined by $${\cal B}= \{f:f\text{ is analytic on $D$ and sup}\{(1-|z|^2)|f'(z)|, z\in D\{<\infty\},$$ where $f'$ is the derivative of $f$; $${\cal B}_0= \{f:f\text{ is analytic on $D$ and }(1-|z|^2)|f'(z)|\to 0\text{ as }|z|\to 1\}.$$ With norm $|.|_{\cal B}$ defined by $|f|_{\cal B}=|f(0)|+ \sup\{(1- |z|^2)|f'(z)|, z\in D\}$, ${\cal B}$ is a Banach space and ${\cal B}_0$ is a closed subspace of ${\cal B}$. If $u$ is an analytic function on $D$ and $\varphi: D\to D$, then the linear weighted composition operator $uC\varphi$ is defined by $$uC\varphi(f)(z)= (uf\circ\varphi)(z)= u(z)f(\varphi(z)).$$ In the results of this paper, the authors derive characterizations for bounded and compact weighted composition operators. In particular, it is shown that (1) $uC\varphi$ is bounded on the Bloch space ${\cal B}$ if and only if (i) $\sup\{(1-|z|^2)|u'(z)|\log(2/(1- |\varphi(z)|^2)), z\in D\{< \infty$; (ii) $\sup\{(1-|z|^2)/(1- |\varphi(z)|^2)\}|u'(z)\varphi'(z)|, z\in D\}<\infty$; and (2) $uC\varphi$ is compact on the Bloch space ${\cal B}$ if and only if expressions of (1) converge to $0$ as $|\varphi(z)|\to 1$; and $uC\varphi$ is compact on ${\cal B}_0$ if and only if the expressions of (1) converge to $0$ as $|z|\to 1$. (3) $uC\varphi$ is bounded on ${\cal B}_0$ if the conditions of (1) are satisfied and $|u(z)\varphi'(z)|(1- |z|^2)\to 0$ as $|z|\to 1$.

47B33Composition operators
47B38Operators on function spaces (general)
47B07Operators defined by compactness properties
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