Products of composition and analytic Toeplitz operators
Let \(X\) be either the (classical) Hardy space \(H^2\) or the Bergman space \(L_a^2\). This is a paper on operators \(X\to X\) of the form \(C_\phi\circ T_\psi\) where, for given analytic functions \(\phi:D\to D\) and \(\psi:D\to{\mathbb C}\), \(C_\phi\) is the composition operator \(f\mapsto f\circ\phi\) and \(T_\psi\) is the multiplication operator \(f\mapsto \psi\cdot f\). It is well-known that \(C_\phi\) always exists and is bounded, but \(T_\psi\) need not exist as a bounded operator. In the present paper, existence, boundedness, and compactness of \(C_\phi\circ T_\psi\) is characterized in terms of Carleson measures and appropriate Nevanlinna counting functions. Moreover, boundedness (compactness) in the Hardy case implies the same for the Bergman case. An example is given to show that \(C_\phi\circ T_\psi\) can be bounded even though \(T_\psi\) is not. Similarly, it may happen that neither \(C_\phi\) nor \(T_\psi\) is compact but \(C_\phi\circ T_\psi\) is compact.
scientific article; zbMATH DE number 1956864