Semigroups of linear operators on \(p\)-Fréchet spaces, \(0<p<1\)
type: Item
instance of: scholarly article
title: Semigroups of linear operators on \(p\)-Fréchet spaces, \(0<p<1\)
zbMATH Open document ID: 1127.47039
published in: Acta Mathematica Hungarica
publication date: 2007-05-08T00:00:00Z
review text: A \(p\)-Fréchet space (sometimes also called a \(p\)-Banach space), \(0<p<1\), is a linear space \((X,+,\| \cdot\| )\) over \(K=\mathbb R\) or \(K=\mathbb C\) which satisfies \(\| x+y\| \leqslant \| x\| +\| y\| \) for all \(x,y\in K\), \(\| x\| =0\) if and only if \(x=0\), \(\| \lambda x\| =| \lambda| ^p\| x\| \) for all \(\lambda \in K\), \(x\in X\), and is complete with respect to the metric \(d(x,y)=\| x-y\| \). Such a space is not necessarily locally compact: consider the Hardy space \(H^p\), the sequence space \(\ell^p\), and the spaces \(L^p\), all with \(0<p<1\). If \(A:X\to X\) is linear and continuous, then \((\sum^m_{j=0}(t^j/j!)A^j(x))_{m=1,2,\dots}\), \(t\in \mathbb R\), \(x\in X\), is a Cauchy sequence, the limit of which defines a semigroup. The authors develop the foundations of a theory of such semigroups and discuss several applications.
reviewed by: Heinrich Hering
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47D09
zbMATH DE Number: 5150249
zbMATH Keywords: \(p\)-Fréchet space \(0<p<1\)
zbMATH Keywords: semigroups of linear operators
zbMATH Keywords: Cauchy problem
author: Sorin G. Gal
author: Jerome A. Goldstein
MaRDI profile type: Publication
full work available at URL: s10474-006-0526-6
OpenAlex ID: W1975060969
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DOI: 10.1007/S10474-006-0526-6
description: scientific article; zbMATH DE number 5150249