A tame splitting theorem for Köthe spaces
type: Item
instance of: scholarly article
title: A tame splitting theorem for Köthe spaces
zbMATH Open document ID: 0639.46001
DOI: 10.1007/BF01198355
author: Kaisa Nyberg
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: Let E and F be Fréchet spaces with fixed gradings \((\mu _ p)\) and \((\nu _ p)\). A linear mapping T:E\(\to F\) is linearly (or multiplicatively) tame if there is a positive integer a and positive numbers \(C_ p\), \(p\in {\mathbb{N}}\), such that \(\nu _ p(Tx)\leq C_ p\mu _{p+a}(x)\) (or \(\nu _ p(Tx)\leq C_ p\mu _{ap}(x))\), for every \(x\in E\) and \(p\in {\mathbb{N}}\). A short exact sequence of Fréchet spaces \(0\to E\to ^{i}X\to ^{q}F\to 0\) is tame if the isomorphisms i:E\(\to i(E)\subset X\) and q:X/i(E)\(\to F\) and their inverses are tame. By definition, a tame short exact sequence splits tamely if q has a tame right inverse or, equivalently, if i has a tame left inverse. Using the method of twisted sums [\textit{T. Ketonen} and \textit{K. Nyberg}: Ann. Acad. Sci. Fenn., Ser. AI 7, 323-335 (1982; Zbl 0531.46003)] a necessary and sufficient condition is given in terms of the gradings of nuclear Köthe spaces E and F under which every tame short exact sequence splits tamely. Concerning the question whether and when tameness implies splitting, this condition is applied to nuclear power series spaces. It is shown that if E is of infinite type and F is of finite type then tameness considerations are irrelevant, while in the other three cases the tameness with respect to certain gradings gives a tame splitting.
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46A45
zbMATH DE Number: 4042532
zbMATH Keywords: Fréchet spaces with fixed gradings
zbMATH Keywords: short exact sequence of Fréchet spaces
zbMATH Keywords: tame short exact sequence
zbMATH Keywords: tame right inverse
zbMATH Keywords: tame left inverse
zbMATH Keywords: twisted sums
zbMATH Keywords: gradings of nuclear Köthe spaces
zbMATH Keywords: nuclear power series spaces
zbMATH Keywords: tame splitting
MaRDI profile type: Publication
cites work: Twisted sums of nuclear Fréchet spaces
cites work: Tameness of Pairs of Nuclear Power Series Spaces and Related Topics
cites work: Splitting twisted sums of nuclear Köthe spaces
cites work: Splitting a twisted sum of Fréchet sequence spaces
cites work: Tame spaces and power series spaces
full work available at URL: https://doi.org/10.1007/bf01198355
OpenAlex ID: W2047533149
description: scientific article; zbMATH DE number 4042532