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Bourin-type inequalities for $τ$-measurable operators in fully symmetric spaces

Teng Zhang

TL;DR

The paper extends Bourin-type inequalities for the crossed Heinz expression $a^t b^{1-t}+b^t a^{1-t}$ to the setting of fully symmetric spaces $E(\tau)$ of $\tau$-measurable operators in semifinite von Neumann algebras. It employs a complex interpolation on a strip to bound an auxiliary expression $f_t=b^{1-t}a^{2t-1}b^{1-t}+a^{1-t}b^{2t-1}a^{1-t}$ and then uses a $2\times2$ block positivity argument to obtain a bridge inequality that controls $b_t=a^t b^{1-t}+b^t a^{1-t}$ by $a+b$. The main result is the sharp bound $\|a^t b^{1-t}+b^t a^{1-t}\|_{E(\tau)} \le 2^{\max\{2|t-1/2|-1/2,0\}} \|a+b\|_{E(\tau)}$ for all $t\in[0,1]$, with the optimal constant 1 on the central range $t\in[1/4,3/4]$, thereby unifying and strengthening prior noncommutative results. This broadens the applicability of Heinz-type inequalities to noncommutative fully symmetric spaces and provides tools for operator inequalities in these spaces, with potential implications for related spectral and interpolation methods.

Abstract

Let $\mathcal{M}\subset B(\mathcal{H})$ be a semifinite von Neumann algebra, where $B(\mathcal{H})$ denotes the algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$, and let $τ$ be a fixed faithful normal semifinite trace on $\mathcal{M}$.Let $E_τ$ be the fully symmetric space associated with a fully symmetric Banach function space $E$ on $[0,\infty)$.Using a complex interpolation argument based on the three-lines theorem on a strip, we show that for positive operators $a,b\in E_τ$ and $t\in[0,1]$, $$ \|a^t b^{1-t}+b^t a^{1-t}\|_{E_τ}\le 2^{\max\{2|t-1/2|-1/2,\;0\}}\;\|a+b\|_{E_τ}. $$ In particular, we obtain the sharp constant $1$ for $t\in[1/4,3/4]$: $$ \|a^t b^{1-t}+b^t a^{1-t}\|_{E_τ}\le \|a+b\|_{E_τ}. $$ This extends the work of Kittaneh--Ricard in \emph{Linear Algebra Appl.} \textbf{710} (2025), 356--362 and covers the results of Liu--He--Zhao in \emph{Acta Math. Sci. Ser. B (Engl. Ed.)} \textbf{46} (2026), 62--68

Bourin-type inequalities for $τ$-measurable operators in fully symmetric spaces

TL;DR

The paper extends Bourin-type inequalities for the crossed Heinz expression to the setting of fully symmetric spaces of -measurable operators in semifinite von Neumann algebras. It employs a complex interpolation on a strip to bound an auxiliary expression and then uses a block positivity argument to obtain a bridge inequality that controls by . The main result is the sharp bound for all , with the optimal constant 1 on the central range , thereby unifying and strengthening prior noncommutative results. This broadens the applicability of Heinz-type inequalities to noncommutative fully symmetric spaces and provides tools for operator inequalities in these spaces, with potential implications for related spectral and interpolation methods.

Abstract

Let be a semifinite von Neumann algebra, where denotes the algebra of all bounded linear operators on a Hilbert space , and let be a fixed faithful normal semifinite trace on .Let be the fully symmetric space associated with a fully symmetric Banach function space on .Using a complex interpolation argument based on the three-lines theorem on a strip, we show that for positive operators and , In particular, we obtain the sharp constant for : This extends the work of Kittaneh--Ricard in \emph{Linear Algebra Appl.} \textbf{710} (2025), 356--362 and covers the results of Liu--He--Zhao in \emph{Acta Math. Sci. Ser. B (Engl. Ed.)} \textbf{46} (2026), 62--68
Paper Structure (12 sections, 17 theorems, 80 equations)

This paper contains 12 sections, 17 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\tau$-measurable operators and generalized singular numbers
  4. Fully symmetric spaces $E(\tau)$
  5. Spectral truncations
  6. Measure topology and continuity tools
  7. Three-lines lemma on a strip
  8. Auxiliary inequalities
  9. Proof of Theorem \ref{['thm:main']}
  10. A complex interpolation estimate for $f_t$
  11. A bridge inequality via $2\times2$ block positivity
  12. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $a,b\ge 0$ in $S(\tau)$ and $z\in S(\tau)$. If $az,zb\in E(\tau)$, then for every $t\in[0,1]$,

Theorems & Definitions (25)

  • Theorem 1.1: Heinz-type inequality in $E(\tau)$, DDSZ20
  • Theorem 1.2
  • Theorem 1.3: Bourin-type bound in $E(\tau)$
  • Lemma 2.1: Unitary invariance
  • Lemma 2.2: Truncations commute with positive powers
  • proof
  • Lemma 2.3: Measure-topology continuity
  • Remark 2.4
  • Lemma 2.5: Fatou-type lower semicontinuity, Pa08
  • Lemma 2.6: Vanishing at infinity when $1\notin E$
  • ...and 15 more