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

作者：Moslehian Mohammad Sal^*; Zhang Fuzhen

来源：Linear Algebra and Its Applications, 2008, 429(8-9): 2159-2167.

DOI：10.1016/j.laa.2008.06.010

• bounded linear operator
• characterization of inner product space
• Hilbert space
• Q-norm
• norm inequality
• schatten p-norm
• continuous filed of operators
• Bouchner integral

Let (sic) be a C*-algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (At)(t epsilon T) be a continuous field of operators in 2I such that the function t 1 -> A(t) is norm continuous on T and the function t 1 -> parallel to A(t)parallel to is integrable. Then the following equality including Bouchner integrals holds
integral T vertical bar A(t)-integral(T)A(s)dP vertical bar(2) dP = integral(T) vertical bar A(t)vertical bar(2) dP - vertical bar integral(T) A(t)dP vertical bar(2).
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.

• 出版日期2008-10-16
• 单位东南大学; 沈阳师范大学