Chebyshev-Type Integral Inequalities for Continuous Fields of Operators Concerning Khatri-Rao Products and Synchronous Properties

摘要

We consider bounded continuous fields of self-adjoint operators which are parametrized by a locally compact Hausdorff space Omega equipped with a finite Radon measure mu. Under certain assumptions on synchronous Khatri-Rao property of the fields of operators, we obtain Chebyshev-type inequalities concerning Khatri-Rao products. We also establish Chebyshev-type inequalities involving Khatri-Rao products and weighted Pythagorean means under certain assumptions of synchronous monotone property of the fields of operators. The Pythagorean means considered here are three classical symmetric means: the geometric mean, the arithmetic mean, and the harmonic mean. Moreover, we derive the Chebyshev-Gruss integral inequality via oscillations when mu is a probability Radon measure. These integral inequalities can be reduced to discrete inequalities by setting Omega to be a finite space equipped with the counting measure. Our results provide analog results for matrices and integrable functions. Furthermore, our results include the results for tensor products of operators, and Khatri-Rao/Kronecker/Hadamard products of matrices, which have been not investigated in the literature.