Operator entropy inequalities

Abstract

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 = (A1, . . . , An) and B = (B1, . . . , Bn) 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

Relative entropy and Tsallis entropy of two accretive operators

“…The Tsallis relative operator entropy is a parametric extension in the sense that lim λ→0 T λ (A|B) = S(A|B). For more details about these operator entropies, we refer the reader to [5] and [11] and the related references cited therein. Our aim in this section is to extend S(A|B) and T λ (A|B) for accretive A, B ∈ B(H).…”

“…For more details about these operator entropies, we refer the reader to [5] and [11] and the related references cited therein. Our aim in this section is to extend S(A|B) and T λ (A|B) for accretive A, B ∈ B(H).…”

Functional Versions of Some Refined and Reversed Operator Mean Inequalities

“…(see, for example, [3]). Various extensions of S(T |S) have been introduced and related operator inequalities have been investigated (see [4][5][6]9]). After [10], the relative entropy can be extended from the case where the variables are positive operators to the case where the variables are (convex) functionals, as follows:…”

Functional Versions for some Operator Entropy Inequalities

“…Since then other extensions of S(A|B) have been introduced and related operator inequalities have been investigated, see [4] and [7] for instance.…”