کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4603971 1631189 2006 12 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Two reverse inequalities associated with Tsallis relative operator entropy via generalized Kantorovich constant and their applications
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
Two reverse inequalities associated with Tsallis relative operator entropy via generalized Kantorovich constant and their applications
چکیده انگلیسی

Recently Tsallis relative operator entropy Tp(A∣B) and Tsallis relative entropy Dp(A∥B) are discussed by Furuichi–Yanagi–Kuriyama. We shall show two reverse inequalities involving Tsallis relative operator entropy Tp(A∣B) via generalized Kantorovich constant K(p). As some applications of two reverse inequalities, we shall show two trace reverse inequalities involving −Tr[Tp(A∣B)] and Dp(A∥B  ) and also a known reverse trace inequality involving the relative operator entropy S^(A|B) by Fujii–Kamei and the Umegaki relative entropy S(A, B) is shown as a simple corollary.We show the following result: Let A and B be strictly positive operators on a Hilbert space H such that M1 I ⩾ A ⩾ m1 I > 0 and M2 I ⩾ B ⩾ m2 I > 0. Put m=m2M1, M=M2m1, h=Mm=M1M2m1m2>1 and p ∈ (0, 1]. Let Φ be normalized positive linear map on B(H). Then the following inequalities hold:equation(i)1-K(p)pΦ(A)♯pΦ(B)+Φ(Tp(A|B))⩾Tp(Φ(A)|Φ(B))⩾Φ(Tp(A|B))and equation(ii)F(p)Φ(A)+Φ(Tp(A|B))⩾Tp(Φ(A)|Φ(B))⩾Φ(Tp(A|B)),F(p)Φ(A)+Φ(Tp(A|B))⩾Tp(Φ(A)|Φ(B))⩾Φ(Tp(A|B)),where K(p) is the generalized Kantorovich constant defined byK(p)=(hp-h)(p-1)(h-1)(p-1)p(hp-1)(hp-h)pand K(p) ∈ (0, 1] and F(p)=mpphp-hh-11-K(p)1p-1⩾0. In addition, let A and B be strictly positive definite matrices, equation(iii)1-K(p)p(Tr[A])1-p(Tr[B])p+Dp(A‖B)⩾-Tr[Tp(A|B)]⩾Dp(A‖B)and equation(iv)F(p)Tr[A]+Dp(A‖B)⩾-Tr[Tp(A|B)]⩾Dp(A‖B).F(p)Tr[A]+Dp(A‖B)⩾-Tr[Tp(A|B)]⩾Dp(A‖B).In particular, both  and  yield the following known result:logS(1)Tr[A]+S(A,B)⩾-Tr[S^(A|B)]⩾S(A,B),where S(1)=h1h-1elogh1h-1 is said to be the Specht ratio and S(1) > 1.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Linear Algebra and its Applications - Volume 412, Issues 2–3, 15 January 2006, Pages 526–537
نویسندگان
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