Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathfrak {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the algebra of bounded linear operators on a complex Hilbert space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. For an operator <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T \in \mathfrak {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="fraktur">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with polar decomposition <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T = U|T|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">|</mo> <mi>T</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>, the generalized <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>-mean transform is defined as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\widehat{T}_\nu (t) = \nu |T|^t U|T|^{1-t} + (1 - \nu )|T|^{1 - t} U |T|^t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="false">^</mo> </mover> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>ν</mi> <mo stretchy="false">|</mo> <mi>T</mi> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> </msup> <msup> <mrow> <mi>U</mi> <mo stretchy="false">|</mo> <mi>T</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mi>t</mi> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mi>T</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mi>t</mi> </mrow> </msup> <mi>U</mi> <msup> <mrow> <mo stretchy="false">|</mo> <mi>T</mi> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\nu , t \in [0, 1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. This transform provides a unified framework encompassing the Aluthge transform, the t-Aluthge transform, and the mean transform. In this paper, we establish several new norm inequalities for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\widehat{T}_\nu (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="false">^</mo> </mover> <mi>ν</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by exploiting the convexity of the function <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$f(\nu ) = \Vert \widehat{T}_\nu (t)\Vert $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo stretchy="false">‖</mo> </mrow> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="false">^</mo> </mover> <mi>ν</mi> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and applying refinements of the Hermite-Hadamard inequality. We provide a complete characterization of operators for which the norm of the generalized <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>-mean transform coincides with the operator norm. Furthermore, we explore the relationship between the powers of the <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>-mean transform and the normaloid property of T. Finally, we present a new asymptotic formula for the spectral radius r(T) in terms of the weighted norms of iterated Aluthge transforms, complementing Gelfand’s classical spectral radius formula.

Norm Inequalities and Characterizations of Normaloid Operators via Generalized Mean Transforms

期刊论文
发表日期：

2026年7月8日

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Salma Aljawi,
Cristian Conde,
Kais Feki,
Hranislav Stanković

摘要

Let $\mathfrak {B}(\mathcal {H})$ $B (H)$ denote the algebra of bounded linear operators on a complex Hilbert space $\mathcal {H}$ $H$ . For an operator $T \in \mathfrak {B}(\mathcal {H})$ $T \in B (H)$ with polar decomposition $T = U|T|$ $T = U | T |$ , the generalized $\nu $ $ν$ -mean transform is defined as $\widehat{T}_\nu (t) = \nu |T|^t U|T|^{1-t} + (1 - \nu )|T|^{1 - t} U |T|^t$ ${\hat{T}}_{ν} (t) = {ν | T |}^{t} {U | T |}^{1 - t} + {(1 - ν) | T |}^{1 - t} U {| T |}^{t}$ , for $\nu , t \in [0, 1]$ $ν, t \in [0, 1]$ . This transform provides a unified framework encompassing the Aluthge transform, the t-Aluthge transform, and the mean transform. In this paper, we establish several new norm inequalities for $\widehat{T}_\nu (t)$ ${\hat{T}}_{ν} (t)$ by exploiting the convexity of the function $f(\nu ) = \Vert \widehat{T}_\nu (t)\Vert $ $f (ν) = ‖ {\hat{T}}_{ν} (t) ‖$ and applying refinements of the Hermite-Hadamard inequality. We provide a complete characterization of operators for which the norm of the generalized $\nu $ $ν$ -mean transform coincides with the operator norm. Furthermore, we explore the relationship between the powers of the $\nu $ $ν$ -mean transform and the normaloid property of T. Finally, we present a new asymptotic formula for the spectral radius r(T) in terms of the weighted norms of iterated Aluthge transforms, complementing Gelfand’s classical spectral radius formula.