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

Shiho Oi, Jyamira Oppekepenguin

TL;DR

This work extends the Koehler–Rosenthal result to general power-bounded operators on complex Banach spaces by leveraging Gelfand's theorem and the Riesz projections. It proves that any invertible power-bounded operator with a finite spectrum decomposes as $T=\sum_{j=1}^m \lambda_j P_j$, where the $P_j$ are the Riesz projections onto the respective eigenspaces, yielding a direct sum decomposition of $X$ and invariant complements. Consequently, such operators are algebraic since $\prod_{j=1}^m (T-\lambda_j I)=0$, and the paper provides a self-contained, detailed argument for the spectral decomposition in the finite-spectrum case. The results have implications for the structure of Banach-space operators with isolated spectral points and connect spectral theory with algebraic operator theory via explicit projections.

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 ${λ_1, λ_2, \dots, λ_m}$, then for every $1 \leq i, j \leq m$, there exist projections $P_j$ on $X$ such that $P_iP_j=δ_{ij}P_i$, $\sum_{j=1}^mP_j=I$, and $T=Σ_{j=1}^m λ_j P_j$. It follows that such an operator $T$ is an algebraic operator.

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

TL;DR

This work extends the Koehler–Rosenthal result to general power-bounded operators on complex Banach spaces by leveraging Gelfand's theorem and the Riesz projections. It proves that any invertible power-bounded operator with a finite spectrum decomposes as , where the are the Riesz projections onto the respective eigenspaces, yielding a direct sum decomposition of and invariant complements. Consequently, such operators are algebraic since , and the paper provides a self-contained, detailed argument for the spectral decomposition in the finite-spectrum case. The results have implications for the structure of Banach-space operators with isolated spectral points and connect spectral theory with algebraic operator theory via explicit projections.

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 is a power-bounded operator on a Banach space whose spectrum consists of finitely many points , then for every , there exist projections on such that , , and . It follows that such an operator is an algebraic operator.
Paper Structure (3 sections, 7 theorems, 35 equations)

This paper contains 3 sections, 7 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. A theorem of Gelfand
  3. A proof of a theorem of Koehler and Rosenthal

Key Result

Lemma 2.1

Let $\chi\colon \mathbb C \to X$ be an entire map. Let $P(\theta)=\sum_{l=k}^mc_le^{il\theta}$ be a triangle polynomial with $c_k\ne 0$. Suppose that there exist real numbers $M>0$, $r_0>0$ and a non-negative integer $\alpha$ such that Then $\chi(w)$ is a polynomial of $w$ with a degree at most $\alpha$. In particular, if $\alpha=0$, then $\chi(w)$ is a constant map.

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3: Satz 1 in gelfand
  • proof
  • Example 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 6 more