Interpolating inequalities for unitarily invariant norms and numerical radii of matrices

Abstract

In this paper, which is a continuation of our works in [9] and [10], we prove several interpolating inequalities for norms and numerical radii of matrices. Special cases of our results present refinements of some known inequalities. Among other results, we prove that if A,B,X are n × n complex matrices such that X is positive semidefinite and t ∈ [0, 1], then

|||AXB∗|||2 ≤  f|_AX1/2|² ,|_BX1/2|^{2 (t)}  

                        ≤ |||X (tA^∗A + (1 − t)B^∗B)|||  |||((1 − t)A^∗A + tB^∗B)X|||

for some special function f and

|||AXB^∗|||²

                        ≤ w_|||·||| ((tA^∗A + (1 − t)B^∗B)X) _w|||·||| (X((1 − t)A^∗A + tB^∗B)) .

Special cases of these two inequalities refine and generalize the well-known Cauchy-Schwarz inequality and the arithmetic-geometric mean inequality for matrices. Here |||·||| and w|||·||| denote any unitarily invariant norm and the generalized numerical radius induced by this norm.

Mathematics Subject Classification (2020): Primary: 15A60; Secondary: 15A18, 15A42, 47A12, 47A30, 47B15.