<p>Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$\{U_{ij}\}_{1\le i<j\le n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>U</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$n\atopwithdelims ()2$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mfrac linethickness="0pt"> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> </math></EquationSource> </InlineEquation> commuting unitaries on a Hilbert space <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {H}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. Suppose <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$U_{ji}:=U^_{ij}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>U</mi> <mrow> <mi mathvariant="italic">ji</mi> </mrow> </msub> <mo>:</mo> <mo>=</mo> <msubsup> <mi>U</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq9.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le i<j\le n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. An <i>n</i>-tuple of power partial isometries <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$(V_1,...,V_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>V</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on Hilbert space <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {H}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> is called <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {U}}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">U</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-twisted power partial isometry with respect to <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq13.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$\{U_{ij}\}_{i<j}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>U</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> (or simply <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {U}}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">U</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-twisted power partial isometry if <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq13.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$\{U_{ij}\}_{i<j}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>U</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is clear from the context) if <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq16.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="615" /> </InlineMediaObject> <EquationSource Format="TEX">$V_i^V_j=U_{ij}V_jV^*_i, V_iV_j=U_{ji}V_jV_i \text {and} V_kU_{ij}=U_{ij}V_k(i,j,k=1,2,...,n,~\text {and}~i\ne j).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>V</mi> <mi>i</mi> <mo>∗</mo> </msubsup> <msub> <mi>V</mi> <mi>j</mi> </msub> <mo>=</mo> <msub> <mi>U</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <msub> <mi>V</mi> <mi>j</mi> </msub> <msubsup> <mi>V</mi> <mi>i</mi> <mo>∗</mo> </msubsup> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <msub> <mi>V</mi> <mi>i</mi> </msub> <msub> <mi>V</mi> <mi>j</mi> </msub> <mo>=</mo> <msub> <mi>U</mi> <mrow> <mi mathvariant="italic">ji</mi> </mrow> </msub> <msub> <mi>V</mi> <mi>j</mi> </msub> <msub> <mi>V</mi> <mi>i</mi> </msub> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mtext>and</mtext> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <msub> <mi>V</mi> <mi>k</mi> </msub> <msub> <mi>U</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>U</mi> <mrow> <mi mathvariant="italic">ij</mi> </mrow> </msub> <msub> <mi>V</mi> <mi>k</mi> </msub> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mspace width="3.33333pt" /> <mtext>and</mtext> <mspace width="3.33333pt" /> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We prove that each <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {U}}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">U</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-twisted power partial isometry admits a Halmos and Wallen (J Math Mech 19:657–663, 1969/1970) type orthogonal decomposition. We provide a concrete model for the decomposition of <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2025_460_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {U}}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">U</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-twisted power partial isometries.</p>

The structure of ${\mathcal {U}}_n$-twisted power partial isometries

期刊论文
发表日期：

2025年7月4日

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Athul Augustine,
P. Shankar

摘要

Let $n>1$ $n > 1$ and let $\{U_{ij}\}_{1\le i<j\le n}$ ${U_{ij}}_{1 \leq i < j \leq n}$ be $n\atopwithdelims ()2$ $(\binom{n}{2})$ commuting unitaries on a Hilbert space ${\mathcal {H}}$ $H$ . Suppose $U_{ji}:=U^*_{ij}$ $U_{ji} : = U_{ij}^{*}$ , $1\le i<j\le n$ $1 \leq i < j \leq n$ . An n-tuple of power partial isometries $(V_1,...,V_n)$ $(V_{1}, . . ., V_{n})$ on Hilbert space ${\mathcal {H}}$ $H$ is called ${\mathcal {U}}_n$ $U_{n}$ -twisted power partial isometry with respect to $\{U_{ij}\}_{i<j}$ ${U_{ij}}_{i < j}$ (or simply ${\mathcal {U}}_n$ $U_{n}$ -twisted power partial isometry if $\{U_{ij}\}_{i<j}$ ${U_{ij}}_{i < j}$ is clear from the context) if $V_i^*V_j=U_{ij}V_jV^*_i, ~~ V_iV_j=U_{ji}V_jV_i ~~\text {and}~~ V_kU_{ij}=U_{ij}V_k~~(i,j,k=1,2,...,n,~\text {and}~i\ne j).$ $V_{i}^{*} V_{j} = U_{ij} V_{j} V_{i}^{*}, V_{i} V_{j} = U_{ji} V_{j} V_{i} and V_{k} U_{ij} = U_{ij} V_{k} (i, j, k = 1, 2, . . ., n, and i \neq j) .$ We prove that each ${\mathcal {U}}_n$ $U_{n}$ -twisted power partial isometry admits a Halmos and Wallen (J Math Mech 19:657–663, 1969/1970) type orthogonal decomposition. We provide a concrete model for the decomposition of ${\mathcal {U}}_n$ $U_{n}$ -twisted power partial isometries.

The structure of \({\mathcal {U}}_n\)-twisted power partial isometries

摘要