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

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• 作者：Mohammad Sal Moslehian ; Fuzhen Zhang
• 关键词：Bounded linear operator ; Characterization of inner product space ; Hilbert space ; Q-Norm ; Norm inequality ; Schatten p-norm ; Continuous filed of operators ; Bouchner integral
• 刊名：Linear Algebra and its Applications
• 出版年：2008
• 出版时间：16 October 2008
• 年：2008
• 卷：429
• 期：8-9
• 页码：2159-2167
• 全文大小：133 K

文摘

Let be a C^*-algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (A_t)_tT be a continuous field of operators in such that the function tA_t is norm continuous on T and the function tA_t is integrable. Then the following equality including Bouchner integrals holds

∫_TA_t-∫_TA_sdP²dP=∫_TA_t²dP-∫_TA_tdP².

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.