Equivalence relations among inequalities for some relative operator entropies
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Positivity
Equivalence relations among inequalities for some relative operator entropies Ismail Nikoufar1
· Maryam Fazlolahi1
Received: 2 February 2020 / Accepted: 17 February 2020 © Springer Nature Switzerland AG 2020
Abstract The relative operator entropy has properties like operator means. In addition, the relative operator entropy has entropy-like properties. In this paper, we prove a Loewner–Heinz type inequality for operator monotone and multiplicative-additive functions and its converse and apply it to the relative operator entropy. We identify the equivalence relations among inequalities for some relative operator entropies in a view point of the perspective. Keywords Operator inequality · Loewner–Heinz inequality · Perspective · Relative operator entropy · Generalized relative operator entropy Mathematics Subject Classification 47A63 · 46L05 · 46L60 · 81T05
1 Introduction and preliminaries Let A and B be two bounded self-adjoint operators on a Hilbert space H. The partial order A ≤ B means that Ax, x ≤ Bx, x for every x ∈ H. A real continuous function f (t) defined on a real interval is said to be operator monotone provided that A ≤ B implies that f (A) ≤ f (B) for any two self-adjoint operators A and B whose spectra are in the domain of f [6]. A non-negative operator monotone function is considered as a variation of an operator mean by the theory of operator means introduced by Kubo and Ando [5]. However, this theory does not include the logarithm and the entropy function which are operator monotone and often used in information theory. So, Fujii [3] introduced the relative operator entropy by generalizing the Kubo– Ando theory. The definition is derived from the Kubo–Ando theory of operator means.
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Ismail Nikoufar [email protected] Maryam Fazlolahi [email protected]
1
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran
123
I. Nikoufar, M. Fazlolahi
The Loewner–Heinz inequality is a fundamental tool for treating operator inequalities. The Loewner–Heinz inequality means that the power function t α is operator monotone on [0, ∞) for 0 < α < 1. Uchiyama [14] showed that A ≤ B if and only if (A+λ)α ≤ (B+λ)α for every λ > 0. He proved a converse of Loewner–Heinz inequality and applied it to the operator mean and spectral order. Moreover, the equivalence relations among inequalities for the geometric and arithmetic means as extensions of a converse of Loewner–Heinz inequality were obtained in [15]. The converse of the Loewner-Heinz inequality in the view point of the perspective and generalized perspective of operator monotone and multiplicative functions was investigated in [11], where perspective inequalities were given equivalent to the Loewner–Heinz inequality. The relative operator entropy has properties like operator means and entropies [3]. In this paper, we obtain the equivalence relations among inequalities for some relative operator entropies in a view point of the perspective [1,8]. We prove a converse of Loewner–Heinz type inequality for operat
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