Imbedding of power series spaces and spaces of analytic functions (Q757782)

Let E denote a nuclear Fréchet space, which satisfies the conditions (DN) and (\(\Omega\)). It is proved that E contains a complemented copy of the power series space \(\Lambda_{\infty}(\epsilon)\) provided the diametral dimensions of E and \(\Lambda_{\infty}(\epsilon)\) are equal and \(\epsilon\) is stable. Several interesting applications of these results to spaces of analytic functions are given. For instance it is shown that the space \({\mathcal O}({\mathbb{C}}^ d)\) of entire functions is isomorphic to a complemented subspace of \({\mathcal O}(D)\), where D is a complete Reinhardt domain, if the characteristic function of D is almost everywhere infinite. In the one dimensional case, the spaces \({\mathcal O}(G)\) and \({\mathcal O}({\mathbb{C}})\) are isomorphic, if and only if their diametral dimensions are equal.