In this paper, our objective is to present the generalized <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2520_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>-mean transform of a Hilbert space operator T, denoted as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2520_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$\widehat{T}_\nu (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="true">^</mo> </mover> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2520_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu , t \in [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. This transform serves as an extension of the generalized mean transform recently introduced by Benhida et al. [Banach J. Math. Anal. 14, 842–855 (2020)], and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2520_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-mean transform studied by Zamani [J. Math. Anal. Appl. 493 (2021)]. Several characterizations involving this new transformation have been established. We have also studied the behaviour of weighted shifts and EP operators under this transform. Among other things, we show that T is an EP operator and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2520_Article_IEq5.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {R}(\widehat{T}_\nu (t))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="true">^</mo> </mover> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is closed if and only if <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2520_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">$\widehat{T}_\nu (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="true">^</mo> </mover> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an EP operator and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2520_Article_IEq7.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="125" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {R}(T)=\mathcal {R}(\widehat{T}_\nu (t))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="true">^</mo> </mover> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The relationship between the numerical radius and the operator norm of the generalized <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2520_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>-mean transform of a Hilbert space operator T, in comparison to those of T itself, is also discussed.

An Extension of the Generalized Mean Transform of Hilbert Space Operators

期刊论文
发表日期：

2025年9月17日

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Najla Altwaijry,
Cristian Conde,
Kais Feki,
Hranislav Stanković

摘要

In this paper, our objective is to present the generalized $\nu $ $ν$ -mean transform of a Hilbert space operator T, denoted as $\widehat{T}_\nu (t)$ ${\hat{T}}_{ν} (t)$ with $\nu , t \in [0,1]$ $ν, t \in [0, 1]$ . This transform serves as an extension of the generalized mean transform recently introduced by Benhida et al. [Banach J. Math. Anal. 14, 842–855 (2020)], and $\lambda $ $λ$ -mean transform studied by Zamani [J. Math. Anal. Appl. 493 (2021)]. Several characterizations involving this new transformation have been established. We have also studied the behaviour of weighted shifts and EP operators under this transform. Among other things, we show that T is an EP operator and $\mathcal {R}(\widehat{T}_\nu (t))$ $R ({\hat{T}}_{ν} (t))$ is closed if and only if $\widehat{T}_\nu (t)$ ${\hat{T}}_{ν} (t)$ is an EP operator and $\mathcal {R}(T)=\mathcal {R}(\widehat{T}_\nu (t))$ $R (T) = R ({\hat{T}}_{ν} (t))$ . The relationship between the numerical radius and the operator norm of the generalized $\nu $ $ν$ -mean transform of a Hilbert space operator T, in comparison to those of T itself, is also discussed.