Weighted composition operators on the Bloch space (Q2721588)

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