Some remarks on Tsallis relative operator entropy

DOI10.1007/S13398-020-00803-9arXiv2001.01342MaRDI QIDQ6332340

Hamid Reza Moradi, Shigeru Furuichi

Publication date: 5 January 2020

Abstract: This paper intends to give some new estimates for Tsallis relative operator entropy $T_{v}left(A|Bight)=fracA{atural}_{v}B-Av$. Let $A$ and $B$ be two positive invertible operators with the spectra contained in the interval $Jsubset(0,infty)$. We prove for any $vinleft[-1,0ight)cupleft(0,1ight]$, ${\displaystyle }$ (ln_v t)A+left( A{{

atural}_{v}}B+tA{{

atural}_{v-1}}B ight)le {{T}_{v}}left( A|B ight) le (ln_v s)A+{{s}^{v-1}}left( B-sA ight) ${\displaystyle }$ where $s,tinJ$. Especially, the upper bound for Tsallis relative operator entropy is a non-trivial new result. Meanwhile, some related and new results are also established. In particular, the monotonicity for Tsallis relative operator entropy is improved. Furthermore, we introduce the exponential type relative operator entropies which are special cases of the perspective and we give inequalities among them and usual relative operator entropies.

Mathematics Subject Classification ID

Linear operator inequalities (47A63) Functional calculus for linear operators (47A60) General theory of (C^*)-algebras (46L05)

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