In this paper, we introduce the class of n-quasi skew [m, C]-symmetric operator on a Hilbert space which is a generalization of skew [m, C] -symmetric operators presented by M. Chō, B. Načevska-Nastovska, and J. Tomiyama. [On skew [m, C]-symmetric operators. Adv. Oper.Theory 2(4), 468–474 (2017)]. An operator T <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\in $</EquationSource> <EquationSource Format="MATHML"><math> <mo>∈</mo> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathscr {B}}({\mathscr {H}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is said to be n-quasi skew [m, C]-symmetric if <Equation ID="Equ15"> <EquationSource Format="TEX">$\begin{aligned} T^{*n}\left( \underset{j=0}{\overset{m}{\sum }}\left( {\begin{array}{c}m\\ j\end{array}}\right) CT^{m-j}CT^{j}\right) T^{n}=0 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi>T</mi> <mrow> <mrow /> <mo>∗</mo> <mi>n</mi> </mrow> </msup> <mfenced close=")" open="("> <munder> <mover> <mo>∑</mo> <mi>m</mi> </mover> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> </munder> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mi>m</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>j</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mi>C</mi> <msup> <mi>T</mi> <mrow> <mi>m</mi> <mo>-</mo> <mi>j</mi> </mrow> </msup> <mi>C</mi> <msup> <mi>T</mi> <mi>j</mi> </msup> </mfenced> <msup> <mi>T</mi> <mi>n</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for some positive integers n and m. Some basic structural properties of this class are established with the help of operator matrix representation. In particular, we study the perturbation of an n-quasi skew [m, C]-symmetric operator with a nilpotent op- erator. Moreover, if T and S are doubly commuting such that T is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-quasi skew [m, C]-symmetric symmetric and S is <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-quasi-<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\left[ k,C\right] $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="]" open="["> <mi>k</mi> <mo>,</mo> <mi>C</mi> </mfenced> </math></EquationSource> </InlineEquation>-symmetric operator, then TS is an <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n_{3}=$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mn>3</mn> </msub> <mo>=</mo> </mrow> </math></EquationSource> </InlineEquation> max<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\left\{ n_{1},n_{2}\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="}" open="{"> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mfenced> </math></EquationSource> </InlineEquation>-quasi skew <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\left[ m+k-1,C \right] $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="]" open="["> <mi>m</mi> <mo>+</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>C</mi> </mfenced> </math></EquationSource> </InlineEquation>-symmetric operator under suitable conditions.

A class of operators related to skew [m, C]-symmetric operators

期刊论文
发表日期：

2026年5月11日

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Imene Djaber,
Messaoud Guesba

摘要

In this paper, we introduce the class of n-quasi skew [m, C]-symmetric operator on a Hilbert space which is a generalization of skew [m, C] -symmetric operators presented by M. Chō, B. Načevska-Nastovska, and J. Tomiyama. [On skew [m, C]-symmetric operators. Adv. Oper.Theory 2(4), 468–474 (2017)]. An operator T $\in $ $\in$ ${\mathscr {B}}({\mathscr {H}})$ $B (H)$ is said to be n-quasi skew [m, C]-symmetric if $\begin{aligned} T^{*n}\left( \underset{j=0}{\overset{m}{\sum }}\left( {\begin{array}{c}m\\ j\end{array}}\right) CT^{m-j}CT^{j}\right) T^{n}=0 \end{aligned}$ $\begin{matrix} T^{* n} (\underset{j = 0}{\sum^{m}}, (\begin{matrix} m \\ j \end{matrix}), C, T^{m - j}, C, T^{j}) T^{n} = 0 \end{matrix}$ for some positive integers n and m. Some basic structural properties of this class are established with the help of operator matrix representation. In particular, we study the perturbation of an n-quasi skew [m, C]-symmetric operator with a nilpotent op- erator. Moreover, if T and S are doubly commuting such that T is $n_{1}$ $n_{1}$ -quasi skew [m, C]-symmetric symmetric and S is $n_{2}$ $n_{2}$ -quasi- $\left[ k,C\right] $ $[k, C]$ -symmetric operator, then TS is an $n_{3}=$ $n_{3} =$ max $\left\{ n_{1},n_{2}\right\} $ $\{n_{1}, n_{2}\}$ -quasi skew $\left[ m+k-1,C \right] $ $[m + k - 1, C]$ -symmetric operator under suitable conditions.