Counterexamples to a multivariable matrix Young conjecture

Abstract

We disprove Conjecture 5.1 of Lin concerning a multivariable extension of Ando's matrix Young inequality for positive semidefinite matrices. For every integer $m\geq4$, the conjectured eigenvalue inequality fails at the largest eigenvalue for $2\times2$ real rank-one orthogonal projections. We also give a $3\times3$ real positive semidefinite counterexample for $m=3$, where the failure occurs at the second eigenvalue.