Spectrum of infinite-dimensional Banach spaces
type: Item
instance of: scholarly article
title: Spectrum of infinite-dimensional Banach spaces
zbMATH Open document ID: 0675.46011
DOI: 10.1007/BF01078587
author: E. V. Tokarev
publication date: 1989-01-01T00:00:00Z
review text: Let X be an infinite dimensional Banach space. Following \textit{L. Schwartz} [Bull. Am. Math. Soc., New Ser. No.4, 135-141 (1982; Zbl 0555.46007)] the author defines the \(\ell_ p\)-spectrum of X as the set S(X) of all those \(p\in [1,\infty]\) for each of which \(\ell_ p\) is finitely represented in X. A subset \(A\subset [1,\infty)\) is called spectral provided \(A=S(X)\) for some X. The author solves a problem posed by L. Schwartz of describing the spectral subsets of [1,\(\infty)\). Namely, a subset A of [1,\(\infty)\) is spectral if and only if it has a regular structure. The latter means that \(A=[p,2]\cup B\) for some compact subset \(B\subset [2,\infty)\) and for some \(p\in [1,2]\).
Mathematics Subject Classification ID: 46B25
zbMATH DE Number: 4105606
zbMATH Keywords: \(\ell _ p\)-spectrum
zbMATH Keywords: spectral subsets of [1,\(\infty )\)
zbMATH Keywords: regular structure
reviewed by: Y. A. Abramovich
MaRDI profile type: Publication
cites work: Geometry and probability in Banach spaces
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4105606