The n-th residual relative operator entropy ${\mathfrak {R}}^{[n]}_{x,y}(A|B)$

Abstract

Let A and B be strictly positive linear operators on a Hilbert space ${\mathcal {H}}$. We have introduced the n-th relative operator entropy $S^{[n]}(A|B)$, the n-th Tsallis relative operator entropy $T_x^{[n]}(A|B)$ and the n-th generalized relative operator entropy $S_y^{[n]}(A|B)$ so far, and have shown their properties. In addition, we have introduced the n-th residual relative operator entropy ${\mathfrak {R}}^{[n]}_{x,y}(A|B)$ which includes $S^{[n]}(A|B)$, $T_x^{[n]}(A|B)$ and $S_y^{[n]}(A|B)$ as special cases (Isa et al. in Ann Funct Anal, 2020). In this paper, we investigate some properties of ${\mathfrak {R}}^{[n]}_{x,y}(A|B)$ and show that they are applicable to see the properties of $S^{[n]}(A|B)$, $T_x^{[n]}(A|B)$ and $S_y^{[n]}(A|B)$.