Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0<\rho \le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>ρ</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$w_{\rho }(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mi>ρ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-operator radius of a Hilbert space operator X. In this paper we present <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>-operator radius characterizations for the product of operators. Among other things, we prove that A is an isometry if and only if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$w_{\rho }(AXA^*) = w_{\rho }(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mi>ρ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>X</mi> <msup> <mi>A</mi> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>w</mi> <mi>ρ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all X. Furthermore, for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$0<\rho \le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>ρ</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\rho \ne 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>≠</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$A=\lambda B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mi>λ</mi> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> is a multiple of a unitary operator for some unit scalar <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$w_{\rho }(AX) = w_{\rho }(XB)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>w</mi> <mi>ρ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>w</mi> <mi>ρ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all X. Some other related results are also discussed.

Remarks on operator radii

期刊论文
发表日期：

2025年8月8日

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Ali Zamani

摘要

Let $0<\rho \le 2$ $0 < ρ \leq 2$ and let $w_{\rho }(X)$ $w_{ρ} (X)$ be the $\rho $ $ρ$ -operator radius of a Hilbert space operator X. In this paper we present $\rho $ $ρ$ -operator radius characterizations for the product of operators. Among other things, we prove that A is an isometry if and only if $w_{\rho }(AXA^*) = w_{\rho }(X)$ $w_{ρ} (A X A^{*}) = w_{ρ} (X)$ for all X. Furthermore, for $0<\rho \le 2$ $0 < ρ \leq 2$ and $\rho \ne 1$ $ρ \neq 1$ , we show that $A=\lambda B$ $A = λ B$ is a multiple of a unitary operator for some unit scalar $\lambda $ $λ$ if and only if $w_{\rho }(AX) = w_{\rho }(XB)$ $w_{ρ} (A X) = w_{ρ} (X B)$ for all X. Some other related results are also discussed.