Schatten p-norm inequalities related to a characterization of inner product spaces

TL;DR

This paper establishes new Schatten p-norm inequalities for operators on Hilbert spaces, providing characterizations of inner product spaces and extending known inequalities with conditions depending on p.

Contribution

It introduces novel Schatten p-norm inequalities involving sums of operators, linking them to inner product space characterizations and extending previous results.

Findings

Derived inequalities for Schatten p-norms with bounds depending on p

Established conditions for inequalities to hold for different p ranges

Connected inequalities to characterizations of inner product spaces

Abstract

Let $A_1, ... A_n$ be operators acting on a separable complex Hilbert space such that $\sum_{i=1}^n A_i=0$. It is shown that if $A_1, ... A_n$ belong to a Schatten $p$-class, for some $p>0$, then 2^{p/2}n^{p-1} \sum_{i=1}^n \|A_i\|^p_p \leq \sum_{i,j=1}^n\|A_i\pm A_j\|^p_p for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. Moreover, \sum_{i,j=1}^n\|A_i\pm A_j\|^2_p \leq 2n^{2/p} \sum_{i=1}^n \|A_i\|^2_p for $0<p\leq 2$, and the reverse inequality holds for $2\leq p<\infty$. These inequalities are related to a characterization of inner product spaces due to E.R. Lorch.