Convolution representation of certain classes of operators connected with multiplication by an analytic function, and their applications (Q2266217)

Let H(G) be the space of all analytic functions in a domain \(G\subseteq C\) which are zero at infinity (if this belongs to G), endowed with the usual topology. Let H'(G) be the dual space of H(G) and L(H(G)) be the algebra of all bounded linear operators in H(G). Assume that an operator \(A\in L(H(G))\) acts in H(G) by the rule \(Ag(z)=\psi (z)g(z)+L(g),\) where \(L\in H'(G)\) and \(\psi\) (z) is a univalent function in G. Applying the method of characteristic functions and the method of convolutions, the author characterizes all commutators \(T\in H(G)\) of A, gives some necessary and sufficient conditions under which such a commutator is an isomorphism, and the similar conditions under which such two commutators are equivalent.