Mathematics – Operator Algebras
Scientific paper
2008-06-16
Linear Alg. Appl. 429 (2008), no. 8-9, 2159-2167
Mathematics
Operator Algebras
10 pages, to appear in Linear Algebra and its Applications
Scientific paper
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.
Moslehian Mohammad Sal
Zhang Fuzhen
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