Non-Schwartzian power series spaces
type: Item
instance of: scholarly article
title: Non-Schwartzian power series spaces
zbMATH Open document ID: 0541.46007
DOI: 10.1007/BF01179750
published in: Mathematische Zeitschrift
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173284
review text: The author presents some isomorphism results for a class of countably normed power series spaces which are not necessarily nuclear or Schwartzian. Let \(a=(a_ n,n\geq 0,a_ n\uparrow \infty)\) be a sequence of numbers and H be a separable Hilbert space. Consider the spaces \(\Lambda_ 0(a;H)=\{x=(x_ k,k\geq 0),x_ k\in H:\| x\|^ 2_ p=\sum_{k}| x_ k|^ 2e^{-2a_ k/p}<\infty,\quad p\geq 1\}\) and \(\Lambda_{\infty}(a;H)\) is of the same structure except that the weights in the sum are \(e^{+2pa_ k}\) now. We can state the main Theorem. The following statements are equivalent: (1) \(\Lambda_ 0(a,H)\) is isomorphic to \(\Lambda_ 0(d,H)\) where \(d=(2^ k,k\geq 0),\) (2) the same as (1) with \(\Lambda_{\infty}\) replacing \(\Lambda_ 0.\) (3) The same as (1) with a satisfying \(\sup_{k}a_{k+1}/a_ k<\infty,\) (4) \(\Lambda_ 0(b,H)\) for some b satisfying \(\sup_{k}b_{k+1}/b_ k<\infty,\) is isomorphic to a closed complemented subspace of \(\Lambda_ 0(a,H).\) (5) The same as (4) with \(\Lambda_{\infty}\) in place of \(\Lambda_ 0.\) Some related consequences are also given.
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 3859738
zbMATH Keywords: Köthe spaces
zbMATH Keywords: hyperlacunary sequences
zbMATH Keywords: nuclear spaces
zbMATH Keywords: countably normed power series spaces
author: Boris Mityagin
reviewed by: Malempati M. Rao
MaRDI profile type: Publication
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OpenAlex ID: W2042914404
description: scientific article; zbMATH DE number 3859738