Composition operators on analytic vector-valued Nevanlinna classes (Q815143)

Let \(\varphi\) be an analytic self map on the unit disc of the complex plane and let \(X\) be a Banach space. This paper investigates composition operators \(C_{\varphi}:f \mapsto f \circ \varphi\) on the vector-valued Nevanlinna classes \(N(X)\) and \(N_a(X)\). In fact, \(C_{\varphi}\) is bounded on both spaces. The main result of the paper shows that \(C_{\varphi}\) is weakly compact on \(N(X)\) if and only if it is weakly compact on the vector-valued Hardy space \(H^1(X)\), and it is weakly compact on \(N_a(X)\) if and only if it is so on the vector-valued Bergman space \(B^1(X)\). Further characterizations of these conditions were obtained by \textit{P.~Liu, F.~Saksman} and \textit{H.--O.\ Tylli} [Pac.\ J.\ Math.\ 184, No.~2, 295--309 (1998; Zbl 0932.47023)], the techniques of which are very important for the proofs of the present paper.