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<p>Motivated by the results of [<CitationRef CitationID="CR7">7</CitationRef>], we introduce generalized quasinormal operators, defined via the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-Aluthge transform. We then characterize this class and establish its fundamental properties, and show that it includes quasinormal operators, idempotents, partial isometries, contractions, cyclic subnormal operators, and generalized quasi-adjoint operators. For <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(w_*\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>w</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msub> </math></EquationSource> </InlineEquation>-hyponormal operators, being generalized quasi-adjoint is equivalent to the same property for their Aluthge transform. Further, some results on the ultraweak closures of elementary operator ranges are also given.</p>

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Generalized Quasinormal Operators via the \(*\)-Aluthge transform I

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Mohamed Morjane,
  • Mohamed Ech-chad

摘要

Motivated by the results of [7], we introduce generalized quasinormal operators, defined via the \(*\) ∗ -Aluthge transform. We then characterize this class and establish its fundamental properties, and show that it includes quasinormal operators, idempotents, partial isometries, contractions, cyclic subnormal operators, and generalized quasi-adjoint operators. For \(w_*\) w ∗ -hyponormal operators, being generalized quasi-adjoint is equivalent to the same property for their Aluthge transform. Further, some results on the ultraweak closures of elementary operator ranges are also given.