Spaces of functions of one variable, analytic in open sets and on compacta

V. P. Zaharyuta

Sbornik. Mathematics, Tome 11 (1970) no. 1, pp. 75-88

Résumé

$A(K)$ is the space of functions analytic on the compactum $K$ of the extended complex plane $\widehat{\mathbf C}$ with the usual locally convex topology; $\overline A_1=A(\{z:|z|\leqslant1\})$, $\overline A_0=\overline A(\{0\})$. The following assertions are proved: 1. For the spaces $A(K)$ and $\overline A_1$ to be isomorphic, it is necessary and sufficient that the set $D =\widehat{\mathbf C}\setminus K$ have no more than a finite number of connected components and that the compactum $K$ be regular (i.e. the Dirichlet problem is solvable in $D$ for any continuous function on $\partial D$). 2. For $A(K)$ and $\overline A_0$ to be isomorphic, it is necessary and sufficient that the logarithmic capacity of the compactum $K$ be equal to zero. 3. For $A(K)$ and $\overline A_0\times\overline A_1$ to be isomorphic, it is necessary and sufficient that the compactum $K$ be represented in the form of the sum of two disjoint nonempty compacta, one of which has zero capacity and the other of which is regular and has a complement consisting of no more than a finite number of connected components. Dual results are obtained for the space $A(D)$, where $D$ is an open set. Bibliography: 20 titles.

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