INTERPOLATING INEQUALITIES FOR FUNCTIONS OF POSITIVE SEMIDEFINITE MATRICES

Let A, B be positive semidefinite nxn matrices, and let alpha is an element of (0, 1). We show that if f is an increasing submultiplicative function on [0, Do) with f (0) = 0 such that f (t) and f(2)(t(1/2)) are convex, then |||f(|AB|)|||(2) <= f(4)(1/(4 alpha(1 alpha))(1/4) (||(alpha f (A) + (1 - alpha) f (B))(2)||| x |||((1 - alpha) f (A) + alpha f (B)(2)|||) for every unitarily invariant norm. Moreover, if alpha is an element of [0, 1] and X is an n x n matrix with X not equal 0, then |||f (|AxB|)|||(2) <= f(||X||)/||X|| |||alpha f(2)(A)X + (1 - alpha)X f(2) (B)||||||(1 - alpha) f(2) (A)X + alpha Xf(2) (B)||| for every unitarily invariant norm. These inequalities present generalizations of recent results of Zou and Jiang and of Audenaert.