算子熵型不等式的一些推广
泛函分析
2017-04-10 v1
摘要
本文利用 Mond-Pe\v{c}ari\'c 方法，在特定条件下建立了一些算子熵不等式的反向不等式。特别地，我们提出：{\tiny \begin{align*} f&\left[\int_T(A_s\natural_{p+1}B_s)d\mu(s)+t_0\left(I_{\mathscr H}-\int_TA_s\natural_pB_sd\mu(s)\right)\right]-\gamma_ff(t_0)\left(I_{\mathscr H}-\int_TA_s\natural_pB_sd\mu(s)\right)\nonumber\ &\le \gamma_f\widetilde{S}_p^f(\mathbf{A}|\mathbf{B})\,, \end{align*}} 其中 T T T μ \mu μ T T T m , M m, M m , M m < 1 < M m<1<M m < 1 < M 0 < m A s ≤ B s ≤ M A s ( s ∈ T ) 0<m A_s \leq B_s \leq M A_s\,\,(s\in T) 0 < m A s ≤ B s ≤ M A s ( s ∈ T ) ∫ T A s = ∫ T B s = I H \int_TA_s=\int_TB_s=I_{\mathscr H} ∫ T A s = ∫ T B s = I H f : ( 0 , ∞ ) → [ 0 , ∞ ) f: (0,\infty) \to [0,\infty) f : ( 0 , ∞ ) → [ 0 , ∞ ) γ f = max { f ( t ) μ f t + ν f : m ≤ t ≤ M , μ f = f ( M ) − f ( m ) M − m , ν f = M f ( m ) − m f ( M ) M − m } \gamma_f=\max\left\{\frac{f(t)}{\mu_f t+\nu_f}: m\leq t\leq M,\mu_f=\frac{f(M)-f(m)}{M-m}, \nu_f=\frac{Mf(m)-mf(M)}{M-m}\right\} γ f = max { μ f t + ν f f ( t ) : m ≤ t ≤ M , μ f = M − m f ( M ) − f ( m ) , ν f = M − m M f ( m ) − m f ( M ) } t 0 ∈ [ m , M ] t_0\in[m,M] t 0 ∈ [ m , M ] p ∈ [ 0 , 1 ] p\in[0,1] p ∈ [ 0 , 1 ] S ~ p f ( A ∣ B ) = ∫ T A s 1 2 ( A s − 1 2 B s A s − 1 2 ) p f ( A s − 1 2 B s A s − 1 2 ) A s 1 2 d μ ( s ) . \widetilde{S}_p^f(\mathbf{A}|\mathbf{B})=\int_TA_s^{\frac{1}{2}}\left(A_s^{-\frac{1}{2}}B_sA_s^{-\frac{1}{2}}\right)^p f\left(A_s^{-\frac{1}{2}}B_sA_s^{-\frac{1}{2}}\right)A_s^{\frac{1}{2}}d\mu(s)\,. S p f ( A ∣ B ) = ∫ T A s 2 1 ( A s − 2 1 B s A s − 2 1 ) p f ( A s − 2 1 B s A s − 2 1 ) A s 2 1 d μ ( s ) .
引用
@article{arxiv.1704.02214,
title = {Some extensions of the operator entropy type inequalities},
author = {Mojtaba Bakherad and Ali Morassaei},
journal= {arXiv preprint arXiv:1704.02214},
year = {2017}
}