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Multiplication operators in spaces of entire functions of finite order and operators of convolution type
O. V. Epifanov
Abstract:
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.
UDC:
517.43+
517.53
MSC: Primary
30D15,
43A22,
44A35; Secondary
30F99,
34A35,
46A12,
46E10,
30C99 Received: 15.04.1982
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