On the continuity of causal operators and the Pták-Stein theorem
type: Item
instance of: scholarly article
title: On the continuity of causal operators and the Pták-Stein theorem
zbMATH Open document ID: 0705.47016
DOI: 10.1007/BF01848125
published in: Periodica Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: Let B be a Banach space and \(\{P^ t\); \(t\in \Lambda \}\) an indexed set of bounded projections of B such that (a) \(\forall r,s\in \Lambda\), \(\exists t\in \Lambda\) so that \(P_ rB\cap P_ sB\supset P_ tB\neq \{0\};\) (b) \(\cap_{t\in \Lambda}P_ tB=\{0\}\); where \(P_ t=I-P_ t.\) A linear operator T (everywhere defined, not necessarily bounded) is called causal with respect to \(\{P^ t\); \(t\in \Lambda \}\), if for any \(t\in \Lambda\), \(P^ tx=P^ ty\) implies \(P^ tTx=P^ tTy\) (x,y\(\in B)\). [also see \textit{A. Feituch} and \textit{R. Saeks}, System theory, New York (1982)]. In the paper under review the automatic continuity of causal operators is discussed. It is shown that (1) if T is causal w.r.t. \(\{P^ t\); \(t\in \Lambda \}\), then there exists \(s\in \Lambda\) such that T is bounded on \(P_ sB;\) (2) if T is causal, translation-invariant on certain Banach spaces of functions on the real line, including Lebesgue spaces, then T is bounded; (3) if \(P_ tB\) has finite codimension for every \(t\in \Lambda\), then every causal operator w.r.t. \(\{P^ t\); \(t\in \Lambda \}\) is bounded. Also, a few examples are given.
Mathematics Subject Classification ID: 47A99
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 4157296
zbMATH Keywords: Pták-Stein theorem
zbMATH Keywords: bounded projections
zbMATH Keywords: automatic continuity of causal operators
author: László Máté
MaRDI profile type: Publication
cites work: Q4093255
cites work: Q3254057
cites work: Q3950367
cites work: Automatic continuity, local type and causality
cites work: A uniform boundedness theorem and mappings into spaces of operators
cites work: Several Theorems on Boundedness and Equicontinuity
description: scientific article; zbMATH DE number 4157296