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

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 $1 \leq i, j \leq m$, there exist projections $P_j$ on $X$ such that $P_iP_j=\delta_{ij}P_i$, $\sum_{j=1}^mP_j=I$, and $T=\Sigma_{j=1}^m \lambda_j P_j$. It follows that such an operator $T$ is an algebraic operator.