Spectral decomposition of power-bounded operators: The finite spectrum case
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| Format: | Preprint |
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2025
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| author | Oi, Shiho Oppekepenguin, Jyamira |
| author_facet | Oi, Shiho Oppekepenguin, Jyamira |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_02769 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral decomposition of power-bounded operators: The finite spectrum case Oi, Shiho Oppekepenguin, Jyamira Functional Analysis 47A10, 47B06, 46B04 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. |
| title | Spectral decomposition of power-bounded operators: The finite spectrum case |
| topic | Functional Analysis 47A10, 47B06, 46B04 |
| url | https://arxiv.org/abs/2501.02769 |