An Operator Equality Involving a Continuous Field of Operators and Its Norm Inequalities
Linear Algebra and its Applications
Bounded linear operator, Characterization of inner product space, Hilbert space, Q-Norm, Norm inequality, Schatten p-norm, Continuous filed of operators, Bouchner integral
Let A be a C∗ -algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (At)t∈T be a continuous field of operators in A such that the function t↦At is norm continuous on T and the function t↦∥At∥ is integrable. Then the following equality including Bouchner integrals holds equation
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 and Zhang, Fuzhen, "An Operator Equality Involving a Continuous Field of Operators and Its Norm Inequalities" (2008). Mathematics Faculty Articles. 27.