An operator extension of the parallelogram law and related norm inequalities

TL;DR

This paper extends the parallelogram law to operators in $C^*$-algebras, providing new inequalities and characterizations of inner product spaces with applications to operator theory.

Contribution

It introduces a general operator parallelogram law, extending classical inequalities like Bohr's inequality to the operator setting.

Findings

Established a new operator parallelogram law.

Derived operator extensions of Bohr's inequality.

Presented several new norm inequalities for operators.

Abstract

We establish a general operator parallelogram law concerning a characterization of inner product spaces, get an operator extension of Bohr's inequality and present several norm inequalities. More precisely, let ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a Radon measure $\mu$ and let $(A_t)_{t\in T}$ be a continuous field of operators in ${\mathfrak A}$ such that the function $t \mapsto A_t$ is norm continuous on $T$ and the function $t \mapsto \|A_t\|$ is integrable. If $\alpha: T \times T \to \mathbb{C}$ is a measurable function such that $\bar{\alpha(t,s)}\alpha(s,t)=1$ for all $t, s \in T$, then we show that \begin{align*} \int_T\int_T&\left|\alpha(t,s) A_t-\alpha(s,t) A_s\right|^2d\mu(t)d\mu(s)+\int_T\int_T\left|\alpha(t,s) B_t-\alpha(s,t) B_s\right|^2d\mu(t)d\mu(s) \nonumber &= 2\int_T\int_T\left|\alpha(t,s) A_t-\alpha(s,t) B_s\right|^2d\mu(t)d\mu(s) - 2\left|\int_T(A_t-B_t)d\mu(t)\right|^2\,. \end{align*}