The n -th residual relative operator entropy $${\mathfrak {R}}^{[n]}_{x,y}(AB)$$
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Advances in Operator Theory https://doi.org/10.1007/s43036-020-00120-3
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ORIGINAL PAPER
The n-th residual relative operator entropy R½n x,y ðAjBÞ Hiroaki Tohyama1 • Eizaburo Kamei2 • Masayuki Watanabe1 Received: 21 August 2020 / Accepted: 30 October 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract Let A and B be strictly positive linear operators on a Hilbert space H. We have introduced the n-th relative operator entropy S½n ðAjBÞ, the n-th Tsallis relative ½n operator entropy Tx ðAjBÞ and the n-th generalized relative operator entropy ½n Sy ðAjBÞ so far, and have shown their properties. In addition, we have introduced ½n the n-th residual relative operator entropy R½n x;y ðAjBÞ which includes S ðAjBÞ, ½n
½n
Tx ðAjBÞ and Sy ðAjBÞ as special cases (Isa et al. in Ann Funct Anal, 2020). In this paper, we investigate some properties of R½n x;y ðAjBÞ and show that they are appli½n
½n
cable to see the properties of S½n ðAjBÞ, Tx ðAjBÞ and Sy ðAjBÞ. Keywords The n-th residual relative operator entropy Tsallis relative operator entropy Generalized relative operator entropy Relative operator entropy
Mathematics Subject Classification 47A63 47A64 94A17
Communicated by Qingxiang Xu. & Hiroaki Tohyama [email protected] Masayuki Watanabe [email protected] 1
Maebashi Institute of Technology, 460-1, Kamisadori, Maebashi, Gunma 371-0816, Japan
2
Kitakaturagi-gun, Nara, Japan
H. Tohyama et al.
1 Introduction A bounded linear operator T on a Hilbert space H is positive (denoted by T 0) if ðTn; nÞ 0 for all n 2 H, and T is said to be strictly positive (denoted by T [ 0) if T is invertible and positive. Throughout this paper, A and B denote positive operators. For strictly positive operators A and B, the relative operator entropy S(A|B), the generalized relative operator entropy Sx ðAjBÞ, the Tsallis relative operator entropy Tx ðAjBÞ are defined as follows [2, 5, 13]: 1 1 1 1 SðAjBÞ A2 log A2 BA2 A2 ; Sx ðAjBÞ ðA \x BÞA1 SðAjBÞ ðx 2 RÞ; A \x B A ðx 2 Rnf0gÞ; Tx ðAjBÞ x T0 ðAjBÞ lim Tx ðAjBÞ ¼ SðAjBÞ ¼ S0 ðAjBÞ; x!0
where A \t B A ðA BA Þt A2 (t 2 R) is a path passing through A and B [1, 3, 10, etc.]. The path A \t B coincides with the weighted geometric operator mean A ]t B if t 2 ½0; 1 (cf. [11]). We remark that B \1t A ¼ A \t B holds. Since the relative operator entropy S(A|B) is given as the derivative of the path A \t B at t ¼ 0, that is, SðAjBÞ ¼ dtd A \t B t¼0 , Fujii et al. [4] gave the viewpoint regard that S(A|B) is the velocity on the path A \t B at t ¼ 0. Similarly, we also d Sx ðAjBÞ as the velocity on A \t B at t ¼ x, that is, Sx ðAjBÞ ¼ dt A \t B t¼x . Based on this viewpoint, we tried to introduce a notion of the acceleration on the path A \t B at t ¼ x which was given as the second derivative of the path at t ¼ x in [8]. According to this perspective, the Tsallis relative operator entropy Tx ðAjBÞ is also seen as the average rate of change of the path A \t B over the interv
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