Applications of amenable semigroups in operator theory
Volume 252 / 2020
Studia Mathematica 252 (2020), 27-48
MSC: Primary 47D03; Secondary 43A07, 47B40, 47A15, 47H10, 46B28.
DOI: 10.4064/sm180408-26-2
Published online: 18 November 2019
Abstract
The paper deals with continuous representations $\mathscr{S} \ni s \mapsto T_s \in \mathscr{L} (E)$ of amenable semigroups $\mathscr{S} $ into the algebra $\mathscr{L} (E)$ of all bounded linear operators on a Banach space $E$. For a closed linear subspace $F$ of $E$, sufficient conditions are given under which there exists a projection $P \in \mathscr{L} (E)$ onto $F$ that commutes with all $T_s$. And when $E$ is a Hilbert space, sufficient conditions are given for the existence of an invertible operator $L \in \mathscr{L} (E)$ such that all $L T_s L^{-1}$ are isometries. Also some results on extending intertwining operators, on renorming and on operators on hereditarily indecomposable Banach spaces are offered.
