The injectivity of the commutant mapping <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma _T$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation> of asymptotically nonvanishing contractions T was considered by G.  P.  Gehér and L.  Kérchy in 2011. In particular, they proved that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma _T$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation> can be injective for contractions with a nontrivial stable space. In this paper it is proved that if a contraction T is such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$I-T^T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>-</mo> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$I-TT^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>-</mo> <mi>T</mi> <msup> <mi>T</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> is of trace class <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathfrak S_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">S</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma _T$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation> is injective if and only if T is of class <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C_{1\cdot }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mrow> <mn>1</mn> <mo>·</mo> </mrow> </msub> </math></EquationSource> </InlineEquation>, that is, the stable space of T is the zero space. Example of a contraction T is given such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$I-T^T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>-</mo> <msup> <mi>T</mi> <mo>∗</mo> </msup> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$I-TT^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <mo>-</mo> <mi>T</mi> <msup> <mi>T</mi> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> are of Schatten–von Neumann class <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathfrak S_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">S</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the stable space of T is not trivial and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\gamma _T$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>T</mi> </msub> </math></EquationSource> </InlineEquation> is injective. Some relationships with normal operators with respect to the injectivity of the commutant mapping are also considered.

Injectivity of the commutant mapping for power bounded operators

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发表日期：

2026年3月25日

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Maria F. Gamal’

摘要

The injectivity of the commutant mapping $\gamma _T$ $γ_{T}$ of asymptotically nonvanishing contractions T was considered by G. P. Gehér and L. Kérchy in 2011. In particular, they proved that $\gamma _T$ $γ_{T}$ can be injective for contractions with a nontrivial stable space. In this paper it is proved that if a contraction T is such that $I-T^*T$ $I - T^{*} T$ or $I-TT^*$ $I - T T^{*}$ is of trace class $\mathfrak S_1$ $S_{1}$ , then $\gamma _T$ $γ_{T}$ is injective if and only if T is of class $C_{1\cdot }$ $C_{1 \cdot}$ , that is, the stable space of T is the zero space. Example of a contraction T is given such that $I-T^*T$ $I - T^{*} T$ and $I-TT^*$ $I - T T^{*}$ are of Schatten–von Neumann class $\mathfrak S_p$ $S_{p}$ for every $p>1$ $p > 1$ , the stable space of T is not trivial and $\gamma _T$ $γ_{T}$ is injective. Some relationships with normal operators with respect to the injectivity of the commutant mapping are also considered.