An operator equality involving a continuous field of operators and its norm inequalities

TL;DR

This paper establishes an operator equality involving continuous fields of operators in a $C^*$-algebra, connecting it to variance and inner product space characterizations, and derives related norm inequalities.

Contribution

It introduces a new operator equality involving Bochner integrals in $C^*$-algebras and applies it to derive norm inequalities, linking operator theory with statistical variance.

Findings

Operator equality involving Bochner integrals in $C^*$-algebras

Derivation of uniform norm inequalities

Derivation of Schatten $p$-norm inequalities

Abstract

Let ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a probability measure $P$ 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. Then the following equality including Bouchner integrals holds \begin{eqnarray}\label{oi} \int_T|A_t - \int_TA_s{\rm d}P|^2 {\rm d}P=\int_T|A_t|^2{\rm d}P - |\int_TA_t{\rm d}P|^2 . \end{eqnarray} This equality is related both to the notion of variance in statistics and to a characterization of inner product spaces. With this operator equality, we present some uniform norm and Schatten $p$-norm inequalities.