Spectral decomposition of power-bounded operators: The finite spectrum case

Abstract

In this paper, we investigate power-bounded operators, including surjective isometries, on Banach spaces. Koehler and Rosenthal asserted that an isolated point in the spectrum of a surjective isometry on a Banach space lies in the point spectrum, with the corresponding eigenspace having an invariant complement. However, they did not provide a detailed proof of this claim, at least as understood by the authors of this manuscript. Here, by applications of a theorem of Gelfand and the Riesz projections, we demonstrate that the theorem of Koehler and Rosenthal holds for any power-bounded operator on a Banach space. This not only furnishes a detailed proof of the theorem but also slightly generalizes its scope. As a result, we establish that if $T:X\to X$ is a power-bounded operator on a Banach space $X$ whose spectrum consists of finitely many points ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m$, then for every $1\le i,j\le m$, there exist projections $P_j$ on $X$ such that $P_i{P}_j={\delta }_{ij}{P}_i$, ${\sum }_{j=1}^m{P}_j=I$, and $T={\Sigma }_{j=1}^m{\lambda }_j{P}_j$. It follows that such an operator $T$ is an algebraic operator.