On the continuity of causal operators and the Pták-Stein theorem
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.
automatic continuity of causal operators
scientific article; zbMATH DE number 4157296