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<p>An operator <i>T</i> on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathcal {H}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> is called <i>pseudo-selfadjoint</i> with <i>S</i> if <i>T</i> is densely defined and there exists a boundedly invertible operator <i>S</i> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathcal {H}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(T^* = STS^{-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mo>=</mo> <mi>S</mi> <mi>T</mi> <msup> <mi>S</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we investigate the distance from a pseudo-selfadjoint operator to the set of selfadjoint operators. We also study some conditions for an operator to be selfadjoint when powers of such an operator are pseudo-selfadjoint. Furthermore, we investigate various local spectral properties of pseudo-selfadjoint operators, including hypercyclicity and weak hypercyclicity. In addition, we show that <i>T</i> is an invertible subnormal operator if and only if <i>T</i> admits a Hermitian factorization of the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(T=AB\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>=</mo> <mi>A</mi> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> for two selfadjoint operators <i>A</i> and <i>B</i>.</p>

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Pseudo-selfadjoint operators and their properties

  • 期刊论文

  • 发表日期：

    2026年7月21日

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  • Muneo Chō,
  • Eungil Ko,
  • Ji Eun Lee

摘要

An operator T on \({\mathcal {H}}\) H is called pseudo-selfadjoint with S if T is densely defined and there exists a boundedly invertible operator S on \({\mathcal {H}}\) H such that \(T^* = STS^{-1}\) T ∗ = S T S - 1 . In this paper, we investigate the distance from a pseudo-selfadjoint operator to the set of selfadjoint operators. We also study some conditions for an operator to be selfadjoint when powers of such an operator are pseudo-selfadjoint. Furthermore, we investigate various local spectral properties of pseudo-selfadjoint operators, including hypercyclicity and weak hypercyclicity. In addition, we show that T is an invertible subnormal operator if and only if T admits a Hermitian factorization of the form \(T=AB\) T = A B for two selfadjoint operators A and B.