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

Abstract

Let $\mathfrak{A}$ be a $C^{\ast }$-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 $t\mapsto \Vert A_t\Vert$ 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.

10 pages, to appear in Linear Algebra and its Applications

References in corpus (1)

• Jensen's Operator Inequality