Unbounded linear operators and nuclear Köthe quotients
type: Item
instance of: scholarly article
title: Unbounded linear operators and nuclear Köthe quotients
zbMATH Open document ID: 0674.46001
DOI: 10.1007/BF01188687
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: A locally convex space F is said to satisfy condition (y) if there is a neighborhood \(V_ 1\) such that every \(u\in F'\) can be expressed as the \(\sigma (F',F)\)-limit of an equicontinuous net \((u_{\alpha})\) where each \(u_{\alpha}\in F'[V^ 0_ 1]\). In particular a Fréchet space F which has the bounded approximation property and admits a continuous norm satisfies (y). The main result of this note is that if there is an unbounded continuous operator T mapping a Fréchet space E into a Fréchet space F which satisfies (y), then there is a nuclear Köthe space \(\lambda\) (A) and a continuous surjection Q: \(F\to \lambda (A)\) such that \(QT(E)=\lambda (A).\) \textit{S. F. Bellenot} and \textit{E. Dubinsky} [Trans. Am. Math. Soc. 273, 579-594 (1982; Zbl 0494.46001)] proved that a separable Fréchet space F has a nuclear Köthe quotient if and only if it satisfies the following condition: there exists k such that for every j there is an i with \[ \sup \{\| U\|^*_ i:\quad u\in F'[U^ 0_ k],\quad \| u\|^*_ j\leq 1\}=\infty \] where \(\|\) \(\|^*_ n\) is the dual norm of the seminorm \(\|\) \(\|_ n\) on F. As one of the simple consequences of the main result the assumption of separability in the theorem stated is removed.
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A20
zbMATH DE Number: 4103714
zbMATH Keywords: locally convex space
zbMATH Keywords: condition (y)
zbMATH Keywords: Frechet space
zbMATH Keywords: bounded approximation property
zbMATH Keywords: unbounded continuous operator
zbMATH Keywords: nuclear Köthe space
zbMATH Keywords: continuous surjection
zbMATH Keywords: nuclear Köthe quotient
author: Süleyman Önal
author: A. Tosun Terz{i}oğlu
MaRDI profile type: Publication
cites work: On a problem of Bellenot and Dubinsky
cites work: Q3918541
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cites work: Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist.
cites work: An example of a nuclear Fréchet space without the bounded approximation property
cites work: On Two Problems of Mityagin
full work available at URL: https://doi.org/10.1007/bf01188687
OpenAlex ID: W1991610827
description: scientific article; zbMATH DE number 4103714