Multiplication operators in spaces of entire functions of finite order and operators of convolution type

O. V. Epifanov

Sbornik. Mathematics, Tome 48 (1984) no. 2, pp. 499-520

Résumé

This article deals with the operator $L_a$ of multiplication by an entire function $a(z)$ with indicator $h(\theta)$ when the order is $\rho$. This operator acts from $[\rho,\mathscr K)$ to $[\rho,\mathscr K+h)$, where $\mathscr K$ is a sequence of indicators, $[\rho,\mathscr K)=\operatorname{span}\bigcup_{k\in\mathscr K}[\rho,k]=\lim_{k\in\mathscr K}\operatorname{ind}[\rho,k]$, with $[\rho,k]$ the standard space of entire functions. It is assumed that the spaces are isomorphic, with respect to a transformation of Borel type, to spaces of functions analytic on many-sheeted closed sets. A criterion is found for the range of $L_a$ to be closed. It is used to derive, in particular, a criterion for an operator of convolution type in a union of $\rho$-convex domains to be an epimorphism, along with known results about convolution operators and operators of convolution type. The conditions connect the directions of non-completely-regular growth of $a(z)$ and of accumulation of its zeros with geometric characteristics of $\mathscr K$. Bibliography: 26 titles.