Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {L}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the algebra of all bounded linear operators on a separable infinite-dimensional complex Hilbert space H. In this paper, we introduce <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\widehat{P}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>P</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation>-symmetric operators <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A=U|A| \in \mathcal {L}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> via the Duggal transform <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\widehat{A}=|A|U$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>A</mi> <mo stretchy="false">^</mo> </mover> <mo>=</mo> <mrow> <mo stretchy="false">|</mo> <mi>A</mi> <mo stretchy="false">|</mo> </mrow> <mi>U</mi> </mrow> </math></EquationSource> </InlineEquation>, as those satisfying <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$AT = TA$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>T</mi> <mo>=</mo> <mi>T</mi> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> implies <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\widehat{A}T = T\widehat{A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>A</mi> <mo stretchy="false">^</mo> </mover> <mi>T</mi> <mo>=</mo> <mi>T</mi> <mover accent="true"> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> for every trace-class operator <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T \in \mathcal {C}_1(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <msub> <mi mathvariant="script">C</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We characterize this class and establish its fundamental properties. It includes quasinormal operators, isometries, co-isometries, partial isometries whose squares are normal, cyclic subnormal operators, and all P-symmetric operators, i.e., operators <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$A \in \mathcal {L}(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$AT = TA$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>T</mi> <mo>=</mo> <mi>T</mi> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> implies <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$A^T = TA^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mo>∗</mo> </msup> <mi>T</mi> <mo>=</mo> <mi>T</mi> <msup> <mi>A</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$T \in \mathcal {C}_1(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <msub> <mi mathvariant="script">C</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. For injective hyponormal operators, P-symmetry is equivalent to that of the Duggal transform. We also provide sufficient conditions for a <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\widehat{P}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>P</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation>-symmetric operator to be normal. The study concludes by presenting results concerning the ultraweak closures of the ranges of inner derivations related to this operator class.

Extension of the notion of P-symmetric operators using the Duggal transform

期刊论文
发表日期：

2026年4月13日

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Elmahdi Sahtouti,
Mohamed Bousmaha,
Mohamed Morjane,
Mohamed Ech-chad

摘要

Let $\mathcal {L}(H)$ $L (H)$ denote the algebra of all bounded linear operators on a separable infinite-dimensional complex Hilbert space H. In this paper, we introduce $\widehat{P}$ $\hat{P}$ -symmetric operators $A=U|A| \in \mathcal {L}(H)$ $A = U | A | \in L (H)$ via the Duggal transform $\widehat{A}=|A|U$ $\hat{A} = | A | U$ , as those satisfying $AT = TA$ $A T = T A$ implies $\widehat{A}T = T\widehat{A}$ $\hat{A} T = T \hat{A}$ for every trace-class operator $T \in \mathcal {C}_1(H)$ $T \in C_{1} (H)$ . We characterize this class and establish its fundamental properties. It includes quasinormal operators, isometries, co-isometries, partial isometries whose squares are normal, cyclic subnormal operators, and all P-symmetric operators, i.e., operators $A \in \mathcal {L}(H)$ $A \in L (H)$ satisfying $AT = TA$ $A T = T A$ implies $A^*T = TA^*$ $A^{*} T = T A^{*}$ for every $T \in \mathcal {C}_1(H)$ $T \in C_{1} (H)$ . For injective hyponormal operators, P-symmetry is equivalent to that of the Duggal transform. We also provide sufficient conditions for a $\widehat{P}$ $\hat{P}$ -symmetric operator to be normal. The study concludes by presenting results concerning the ultraweak closures of the ranges of inner derivations related to this operator class.