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Some remarks on Tsallis relative operator entropy

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Abstract

This paper intends to give some new estimates for Tsallis relative operator entropy \({{T}_{v}}\left( A|B \right) =\frac{A{{\natural }_{v}}B-A}{v}\). Let A and B be two positive invertible operators with the spectra contained in the interval \(J \subset (0,\infty )\). We prove for any \(v\in \left[ -1,0 \right) \cup \left( 0,1 \right] \),

$$\begin{aligned} (\ln _v t)A+\left( A{{\natural }_{v}}B+tA{{\natural }_{v-1}}B \right) \le {{T}_{v}}\left( A|B \right) \le (\ln _v s)A+{{s}^{v-1}}\left( B-sA \right) \end{aligned}$$

where \(s,t\in J\). 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.

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Acknowledgements

The author (S.F.) was partially supported by JSPS KAKENHI Grant Number 16K05257.

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Correspondence to Shigeru Furuichi.

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Furuichi, S., Moradi, H.R. Some remarks on Tsallis relative operator entropy. RACSAM 114, 69 (2020) doi:10.1007/s13398-020-00803-9

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Keywords

  • Relative operator entropy
  • Tsallis relative operator entropy
  • Operator inequality
  • Refined Young inequality

Mathematics Subject Classification

  • Primary 47A63
  • Secondary 46L05
  • 47A60