OPERATOR ENTROPY INEQUALITIES

We investigate a notion of relative operator entropy, which develops the theory started by J. I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341-348]. For two finite sequences A = (A(1), ... , A(n)) and B = (B-1, ... , B-n) of positive operators acting on a Hilbert space, a real number q and an operator monotone function f we extend the concept of entropy by setting S-q(f)(A vertical bar B) := Sigma(n)(j=1) A(j)(1/2)(A(j)(-1/2)B(j)A(j)(-1/2))(q) f(A(j)(-1/2)B(j)A(j)(-1/2))A(j)(1/2), and then give upper and lower bounds for S-q(f)(A vertical bar B) as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219-235] under certain conditions. As an application, some inequalities concerning the classical Shannon entropy are deduced.