Multirectangular invariants for power Köthe spaces
type: Item
instance of: scholarly article
title: Multirectangular invariants for power Köthe spaces
zbMATH Open document ID: 1077.46003
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-10-12T00:00:00Z
review text: This paper investigates the isomorphic classification of power Köthe spaces of first type \(E(\lambda,a)\) which are defined as follows: For sequences of positive numbers \(a=(a_i)_i\) and \(\lambda=(\lambda_i)_i\), \(E(\lambda,a)\) is the Köthe echelon space defined by the Köthe matrix \(A=(a_p)_p,\) \(a_p=(a_{i,p})_i\), \(a_{i,p}= \exp((-1/p + \lambda_i p)a_i)\), for each natural number \(i,p\). The class of all such spaces is the smallest class of Köthe echelon spaces, containing all Cartesian and projective tensor products of power series spaces and closed with respect to subspaces spanned by a subset of the canonical basis. Continuing previous work by Djakov and the present authors, the following result proven here solves an open problem in the area: Let \(X=E(\lambda,a)\) and \(Y=E(\mu,b)\) be two power Köthe spaces of first type which are isomorphic. If \(L\) is a basic subspace of \(X\) which is isomorphic to a power series space of finite (resp.\ infinite) type, then there is a basic subspace \(M\) of \(Y\) of the same kind which is quasidiagonally isomorphic to \(L\). Two Köthe echelon spaces \(\lambda(A)\) and \(\lambda(B)\) are quasidiagonally isomorphic if there is an isomorphism \(T\) from \(\lambda(A)\) onto \(\lambda(B)\) such that \(T e_i=t_i e_{\sigma(i)}\) for each \(i\) and for a sequence of scalars \((t_i)_i\) and a bijection \(\sigma\) on the natural numbers. The proof is long and technical. The crucial tools are improvements of the compound invariants which had been investigated before by the authors.
reviewed by: José Bonet
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A63
zbMATH DE Number: 2107215
zbMATH Keywords: linear topological invariants
zbMATH Keywords: compound invariants
zbMATH Keywords: quasidiagonal isomorphism
zbMATH Keywords: Köthe spaces
zbMATH Keywords: power series spaces
zbMATH Keywords: \(m\)-rectangular characteristics
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.04.034
OpenAlex ID: W1988388560
cites work: Q4894764
cites work: Q3265048
cites work: Q3925331
cites work: Q4894768
cites work: Compound invariants and embeddings of Cartesian products
cites work: Compound Invariants and Mixed F-, DF-Power Spaces
cites work: Q4218443
cites work: Q4355939
cites work: Q4894769
cites work: Every nuclear Fréchet space with a regular basis has the quasi-equivalence property
cites work: A short proof of the theorem of Crone and Robinson on quasi-equivalence of regular bases
cites work: Q3275056
cites work: Q5521905
cites work: Q3219294
cites work: Q5518756
cites work: Q3248223
cites work: Q4130523
cites work: Q3965130
cites work: Q4340900
cites work: APPROXIMATE DIMENSION AND BASES IN NUCLEAR SPACES
cites work: Q5617989
cites work: The structure of subspaces of an infinite Hilbert scale
cites work: Non-Schwartzian power series spaces
cites work: Q3245480
cites work: Q3350159
cites work: Q5535819
cites work: Q5574974
cites work: Isomorphisms of Cartesian products of linear topological spaces
cites work: On the isomorphism of cartesian products of locally convex spaces
cites work: Q4080183
cites work: Q5687894
DOI: 10.1016/J.JMAA.2004.04.034
Recommended article: Multirectangular characteristic invariants for power \(\ell\)-Köthe spaces of first type
Recommended article: Q4218443
Recommended article: Multirectangular characteristics for Köthe power spaces of the second kind
Recommended article: Q4952923
Recommended article: Power invariants of certain point sets
Recommended article: Geometric invariants of locally symmetric spaces and multiplier theorems
Recommended article: Q5687421
Recommended article: The invariant subspace problem for non-Archimedean Köthe spaces
Recommended article: Birkhoff-Kellogg theorems on invariant directions for multimaps
Recommended article: Invariant subspaces on multiply connected domains
description: scientific article; zbMATH DE number 2107215