Operator-valued versions of matrix-norm inequalities

Jameson, Graham (2019) Operator-valued versions of matrix-norm inequalities. American Mathematical Monthly, 126 (9). pp. 809-815. ISSN 0002-9890

Text (opineq.amm)
opineq.amm.pdf - Accepted Version
Available under License Creative Commons Attribution-NonCommercial.
Download (154kB)

Abstract

We describe a rather striking extension of a wide class of inequalities. Some famous classical inequalities, such as those of Hardy and Hilbert, equate to the evaluation of the norm of a matrix operator. Such inequalities can be presented in two versions, linear and bilinear. We show that in all such inequalities, the scalars can be replaced by operators on a Hilbert space, with the conclusions taking the form of an operator inequality in the usual sense. With careful formulation, a similar extension applies to the Cauchy–Schwarz inequality.