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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_441_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$T: X \rightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> is a power-bounded operator on a Banach space X whose spectrum consists of finitely many points <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_441_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">${\lambda _1, \lambda _2, \dots , \lambda _m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>λ</mi> <mi>m</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, then for every <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_441_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$1 \le i, j \le m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≤</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, there exist projections <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_441_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$P_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> on X such that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_441_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$P_iP_j=\delta _{ij}P_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> <msub> <mi>P</mi> <mi>j</mi> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_441_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum _{j=1}^mP_j=I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <msub> <mi>P</mi> <mi>j</mi> </msub> <mo>=</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_441_Article_IEq7.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$T=\Sigma _{j=1}^m \lambda _j P_j$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>=</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <msub> <mi>λ</mi> <mi>j</mi> </msub> <msub> <mi>P</mi> <mi>j</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. It follows that such an operator T is an algebraic operator.

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

期刊论文
发表日期：

2025年5月12日

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Shiho Oi,
Jyamira Oppekepenguin

摘要

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 $T: X \rightarrow X$ $T : X \to X$ is a power-bounded operator on a Banach space X whose spectrum consists of finitely many points ${\lambda _1, \lambda _2, \dots , \lambda _m}$ $λ_{1}, λ_{2}, \dots, λ_{m}$ , then for every $1 \le i, j \le m$ $1 \leq i, j \leq m$ , there exist projections $P_j$ $P_{j}$ on X such that $P_iP_j=\delta _{ij}P_i$ $P_{i} P_{j} = δ_{ij} P_{i}$ , $\sum _{j=1}^mP_j=I$ $\sum_{j = 1}^{m} P_{j} = I$ , and $T=\Sigma _{j=1}^m \lambda _j P_j$ $T = Σ_{j = 1}^{m} λ_{j} P_{j}$ . It follows that such an operator T is an algebraic operator.