A class of operators on weighted Bergman spaces (Q1030484)

For an analytic function \(\phi(z)=\sum_{m=0}^\infty\phi_m z^m\) defined on the unit disc, the authors introduce the operator \(V_\phi(f)(z)=\int_{\mathbb D}\frac{\phi(w)f(\bar w z)}{(1-\bar w z)^{2+\alpha}}\, dA_\alpha(w)\) acting on functions \(f\in A^\alpha\), where \(A^\alpha\) stands for the weighted Bergman space of those functions analytic in the disc belonging to \(L^2(dA_\alpha)\) with \(dA_\alpha=(\alpha+1)(1-|z|^2)^\alpha \,dA(z)\). The authors use the matrix representation of \(V_\phi\), given by \(V_\phi(m,n)=A_{n,m}\phi_m\) for \(m\geq n\) and \(V_\phi(m,n)=0\) otherwise, to show that the operator \(V_\phi\) is bounded, compact or belongs to the Schatten class \(S_p\) if and only if the sequence \(\Big((2^{n}\sum_{k\in I_n}|\phi_k|^2)^{1/2}\Big)_n\), where \(I_n= [2^n-2, 2^{n+1}-3]\cap \mathbb N\), belongs to \(\ell^\infty\), \(c_0\) or \(\ell^p\).