For a positive operator A on an infinite-dimensional Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n,\,m\in \mathbb {N}^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>m</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, an operator <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T\in \mathcal {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <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 (A, n)-symmetric if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\displaystyle \sum _{k=0}^{n}(-1)^{n-k} \;{n\atopwithdelims ()k}\;T^{k}AT^{n-k}=0$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>n</mi> </munderover> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mi>k</mi> </mrow> </msup> <mspace width="0.277778em" /> <mfenced close=")" open="("> <mfrac linethickness="0pt"> <mi>n</mi> <mi>k</mi> </mfrac> </mfenced> <mspace width="0.277778em" /> <msup> <mi>T</mi> <mrow> <mrow /> <mo>∗</mo> <mi>k</mi> </mrow> </msup> <mi>A</mi> <msup> <mi>T</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mstyle> </math></EquationSource> </InlineEquation> and exponentially (A, m)-isometric if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \displaystyle \sum _{k=0}^{m}(-1)^{m-k} \;{m\atopwithdelims ()k}\;e^{kT^*} Ae^{kT}=0.$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>m</mi> </munderover> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mi>k</mi> </mrow> </msup> <mspace width="0.277778em" /> <mfenced close=")" open="("> <mfrac linethickness="0pt"> <mi>m</mi> <mi>k</mi> </mfrac> </mfenced> <mspace width="0.277778em" /> <msup> <mi>e</mi> <mrow> <mi>k</mi> <msup> <mi>T</mi> <mo>∗</mo> </msup> </mrow> </msup> <mi>A</mi> <msup> <mi>e</mi> <mrow> <mi mathvariant="italic">kT</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </mstyle> </math></EquationSource> </InlineEquation> These classes of operators seem a natural generalizations of m-symmetries and exponentially m-isometries on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> introduced by Helton, J. ( [<CitationRef CitationID="CR20">20</CitationRef>, <CitationRef CitationID="CR21">21</CitationRef>]) and Hedayatian, K. ( [<CitationRef CitationID="CR26">26</CitationRef>]), respectively. This paper concerns the study of some connection between (A, n)-symmetries, exponentially (A, m)-isometries and related families. We collect some of their interesting properties. The dynamics of such a families will be considered. We will show that, under suitable assumptions on A and T, there is no N-supercyclic (A, n)-symmetric and exponentially (A, m)-isometric operators on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>.

Classes of operators related to (A, n)-symmetries and (A, m)-isometries

期刊论文
发表日期：

2026年5月9日

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Wided Benmassaoud,
Rchid Rabaoui

摘要

For a positive operator A on an infinite-dimensional Hilbert space $\mathcal {H}$ $H$ and $n,\,m\in \mathbb {N}^*$ $n, m \in N^{*}$ , an operator $T\in \mathcal {B}(\mathcal {H})$ $T \in B (H)$ is said to be (A, n)-symmetric if $\displaystyle \sum _{k=0}^{n}(-1)^{n-k} \;{n\atopwithdelims ()k}\;T^{*k}AT^{n-k}=0$ $\sum_{k = 0}^{n} {(- 1)}^{n - k} (\binom{n}{k}) T^{* k} A T^{n - k} = 0$ and exponentially (A, m)-isometric if $ \displaystyle \sum _{k=0}^{m}(-1)^{m-k} \;{m\atopwithdelims ()k}\;e^{kT^*} Ae^{kT}=0.$ $\sum_{k = 0}^{m} {(- 1)}^{m - k} (\binom{m}{k}) e^{k T^{*}} A e^{kT} = 0 .$ These classes of operators seem a natural generalizations of m-symmetries and exponentially m-isometries on $\mathcal {H}$ $H$ introduced by Helton, J. ( [20, 21]) and Hedayatian, K. ( [26]), respectively. This paper concerns the study of some connection between (A, n)-symmetries, exponentially (A, m)-isometries and related families. We collect some of their interesting properties. The dynamics of such a families will be considered. We will show that, under suitable assumptions on A and T, there is no N-supercyclic (A, n)-symmetric and exponentially (A, m)-isometric operators on $\mathcal {H}$ $H$ .