Suppose Q is an idempotent operator. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Q=V_Q|Q|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo>=</mo> <msub> <mi>V</mi> <mi>Q</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mi>Q</mi> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Psi =U_\Psi |\Psi |$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <msub> <mi>U</mi> <mi mathvariant="normal">Ψ</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the polar decompositions of Q and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Psi =2Q-I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mn>2</mn> <mi>Q</mi> <mo>-</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation>, respectively. Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi =Q+Q^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mi>Q</mi> <mo>+</mo> <msup> <mi>Q</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Upsilon =Q+Q^-I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Υ</mi> <mo>=</mo> <mi>Q</mi> <mo>+</mo> <msup> <mi>Q</mi> <mo>∗</mo> </msup> <mo>-</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation>. We prove that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Q=|Q^||Q|=\frac{1}{2}\big [(2|\Phi |-|\Psi |)U_{\Psi } -I\big ],$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi>Q</mi> <mo>=</mo> <mo stretchy="false">|</mo> </mrow> <msup> <mi>Q</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> <mi>Q</mi> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">[</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">|</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">|</mo> </mrow> <mo>-</mo> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <msub> <mi>U</mi> <mi mathvariant="normal">Ψ</mi> </msub> <mo>-</mo> <mi>I</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">]</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$|\Phi |=|Q|+|Q^|= |\Upsilon |+ U_{\Psi },$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">|</mo> <mi>Q</mi> <mo stretchy="false">|</mo> <mo>+</mo> <mo stretchy="false">|</mo> </mrow> <msup> <mi>Q</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">|</mo> <mo>=</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">Υ</mi> <mo stretchy="false">|</mo> <mo>+</mo> </mrow> <msub> <mi>U</mi> <mi mathvariant="normal">Ψ</mi> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ |Q|-|Q^|=\frac{1}{2}(|\Psi |-|\Psi ^|)=|\Psi |-|\Upsilon |=|\Upsilon |-|\Psi ^|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>Q</mi> <mo stretchy="false">|</mo> <mo>-</mo> <mo stretchy="false">|</mo> </mrow> <msup> <mi>Q</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">|</mo> <mo>=</mo> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">|</mo> <mo>-</mo> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Ψ</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">|</mo> <mo>-</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">Υ</mi> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Υ</mi> <mo stretchy="false">|</mo> <mo>-</mo> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Ψ</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$U_{\Psi }= U^_{\Psi }= U^{-1}_{\Psi }=|\Psi |\Psi =|\Psi ^|\Psi ^=\Psi |\Psi ^| =\Psi ^|\Psi |=|\Upsilon |\Upsilon ^{-1}=|\Upsilon |^{-1}\Upsilon .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>U</mi> <mi mathvariant="normal">Ψ</mi> </msub> <mo>=</mo> <msubsup> <mi>U</mi> <mi mathvariant="normal">Ψ</mi> <mo>∗</mo> </msubsup> <mo>=</mo> <msubsup> <mi>U</mi> <mi mathvariant="normal">Ψ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>=</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Ψ</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Ψ</mi> <mo>∗</mo> </msup> <mrow> <mo>=</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Ψ</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">|</mo> <mo>=</mo> </mrow> <msup> <mi mathvariant="normal">Ψ</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Υ</mi> <mo stretchy="false">|</mo> </mrow> <msup> <mi mathvariant="normal">Υ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">Υ</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi mathvariant="normal">Υ</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The equivalent conditions for positive operators A and B which can be written as <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$A=Q^Q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <msup> <mi>Q</mi> <mo>∗</mo> </msup> <mi>Q</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$B=QQ^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>=</mo> <mi>Q</mi> <msup> <mi>Q</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> are obtained. Also, we characterize the idempotents Q, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$Q_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$Q_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq14"> <EquationSource Format="TEX">${\mathcal {R}}(Q) \subseteq {\mathcal {R}}(Q_1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊆</mo> <mi mathvariant="script">R</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$ {\mathcal {N}}(Q_2) \subseteq {\mathcal {N}}(Q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">N</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>⊆</mo> <mi mathvariant="script">N</mi> <mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In particular, the equivalent condition for idempotent Q which can be written as the product <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$Q=Q_1Q_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo>=</mo> <msub> <mi>Q</mi> <mn>1</mn> </msub> <msub> <mi>Q</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is described.

On the idempotent operator and polar decomposition

期刊论文
发表日期：

2026年4月13日

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Chunyuan Deng,
Xingxue Fu,
Xiaohui Li

摘要

Suppose Q is an idempotent operator. Let $Q=V_Q|Q|$ $Q = V_{Q} | Q |$ and $\Psi =U_\Psi |\Psi |$ $Ψ = U_{Ψ} | Ψ |$ be the polar decompositions of Q and $\Psi =2Q-I$ $Ψ = 2 Q - I$ , respectively. Let $\Phi =Q+Q^*$ $Φ = Q + Q^{*}$ and $\Upsilon =Q+Q^*-I$ $Υ = Q + Q^{*} - I$ . We prove that $Q=|Q^*||Q|=\frac{1}{2}\big [(2|\Phi |-|\Psi |)U_{\Psi } -I\big ],$ $Q = | Q^{*} | | Q | = \frac{1}{2} [(2 | Φ | - | Ψ |) U_{Ψ} - I],$ $|\Phi |=|Q|+|Q^*|= |\Upsilon |+ U_{\Psi },$ $| Φ | = | Q | + | Q^{*} | = | Υ | + U_{Ψ},$ $ |Q|-|Q^*|=\frac{1}{2}(|\Psi |-|\Psi ^*|)=|\Psi |-|\Upsilon |=|\Upsilon |-|\Psi ^*|$ $| Q | - | Q^{*} | = \frac{1}{2} (| Ψ | - | Ψ^{*} |) = | Ψ | - | Υ | = | Υ | - | Ψ^{*} |$ and $U_{\Psi }= U^*_{\Psi }= U^{-1}_{\Psi }=|\Psi |\Psi =|\Psi ^*|\Psi ^*=\Psi |\Psi ^*| =\Psi ^*|\Psi |=|\Upsilon |\Upsilon ^{-1}=|\Upsilon |^{-1}\Upsilon .$ $U_{Ψ} = U_{Ψ}^{*} = U_{Ψ}^{- 1} = | Ψ | Ψ = | Ψ^{*} | Ψ^{*} = Ψ | Ψ^{*} | = Ψ^{*} | Ψ | = | Υ | Υ^{- 1} = {| Υ |}^{- 1} Υ .$ The equivalent conditions for positive operators A and B which can be written as $A=Q^*Q$ $A = Q^{*} Q$ and $B=QQ^*$ $B = Q Q^{*}$ are obtained. Also, we characterize the idempotents Q, $Q_1$ $Q_{1}$ and $Q_2$ $Q_{2}$ such that ${\mathcal {R}}(Q) \subseteq {\mathcal {R}}(Q_1)$ $R (Q) \subseteq R (Q_{1})$ or $ {\mathcal {N}}(Q_2) \subseteq {\mathcal {N}}(Q)$ $N (Q_{2}) \subseteq N (Q)$ . In particular, the equivalent condition for idempotent Q which can be written as the product $Q=Q_1Q_2$ $Q = Q_{1} Q_{2}$ is described.