Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$T = A + iB$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mi>i</mi> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> be the Cartesian decomposition of an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \times n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> matrix T, where A and B are Hermitian. Bhatia and Zhan proved that if A is positive semidefinite, we have <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\left\| T\right\| ^2_p \ge \left\| A\right\| ^2_p + 2^{1 - \nicefrac {2}{p}}\left\| B\right\| ^2_p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mfenced close="∥" open="∥"> <mi>T</mi> </mfenced> <mi>p</mi> <mn>2</mn> </msubsup> <mo>≥</mo> <msubsup> <mfenced close="∥" open="∥"> <mi>A</mi> </mfenced> <mi>p</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mn>2</mn> <mrow> <mn>1</mn> <mo>-</mo> <mfrac bevelled="true"> <mn>2</mn> <mi>p</mi> </mfrac> </mrow> </msup> <msubsup> <mfenced close="∥" open="∥"> <mi>B</mi> </mfenced> <mi>p</mi> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> for every Schatten-p norm for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1 <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We show that this inequality is sharp, settling a question left open by Bhatia and Zhan.

Cartesian Decomposition and Schatten Norms: On the Sharpness of Some Inequalities of Bhatia and Zhan

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2026年5月16日

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Pritam Chandra

摘要

Let $T = A + iB$ $T = A + i B$ be the Cartesian decomposition of an $n \times n$ $n \times n$ matrix T, where A and B are Hermitian. Bhatia and Zhan proved that if A is positive semidefinite, we have $\left\| T\right\| ^2_p \ge \left\| A\right\| ^2_p + 2^{1 - \nicefrac {2}{p}}\left\| B\right\| ^2_p$ ${∥T∥}_{p}^{2} \geq {∥A∥}_{p}^{2} + 2^{1 - \frac{2}{p}} {∥B∥}_{p}^{2}$ for every Schatten-p norm for $1< p < 2$ $1 < p < 2$ . We show that this inequality is sharp, settling a question left open by Bhatia and Zhan.