Refined Heinz–Kato–Loewner inequalities

  • Stefan Steinerberger

    Yale University, New Haven, USA
Refined Heinz–Kato–Loewner inequalities cover

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Abstract

A version of the Cauchy–Schwarz inequality in operator theory is the following: for any two symmetric, positive definite matrices and arbitrary

This inequality is classical and equivalent to the celebrated Heinz–Löwner, Heinz–Kato and Cordes inequalities. We characterize cases of equality: in particular, after factoring out the symmetry coming from multiplication with scalars , the case of equality requires that and have a common eigenvalue . We also derive improved estimates and show that if either or does not have a solution, i.e. if where

then there is an improved inequality

for some that only depends only on and . We obtain similar results for the McIntosh inequality and the Cordes inequality and expect the method to have many further applications.

Cite this article

Stefan Steinerberger, Refined Heinz–Kato–Loewner inequalities. J. Spectr. Theory 9 (2019), no. 1, pp. 1–20

DOI 10.4171/JST/239