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

Mohammad Sal Moslehian and Fuzhen Zhang

Arxiv ID: 0806.2633•Last updated: 7/23/2021

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.

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