Title: Matrix Young inequalities for the Hilbert–Schmidt norm
Abstract: Let A,B, and X be n×n complex matrices such that A and B are positive semidefinite. If p,q>1 with 1p+1q=1, it is shown that ∥1pApX+1qXBq∥22⩾1r2ApX−XBq22+AXB22, where r=max(p,q) and ·2 is the Hilbert–Schmidt norm. Generalizations and applications of this inequality are also considered.