论文标题
与操作员单调功能有关的规范不平等现象
Norm inequalities related to operator monotone functions
论文作者
论文摘要
让$ a $是希尔伯特太空$ h $的正定运算符，$ |||。||| $是$ b（h）$的单位不变的规范。我们表明，如果$ f $是$（0，\ infty）$和$ n \ in \ mathbb {n} $上的算法单调功能，则$ |||| d^n f（a）||| \ leq \ leq \ | f^{（n）}（n）}（n）}（a）}（a） $ b（h）$中的所有正确定运营商的集合。我们建立了一些涉及可区分功能的某些HERMITE-HADAMARD类型不平等的右侧估计值，并且它们在自我伴随操作员集合中引起的地图的规范是凸，Quasi-convex或$ s $ convex。 作为应用程序，我们获得了$ ||| f（b）-f（a）||| $在$ ||| b-a ||| $中的一些界限。例如，让$ f，g $为$（0，\ infty）$上的两个操作员单调函数。然后，对于每个单位不变的规范$ |||。||| $和每个积极的确定运算符$ a，b $，b $ ＆\ leq || b-a ||| \ big [\ max \ left \ {\ | f'（a）\ |，\ | f'（b）\ | \ | \ | \ right \} \ times \ times \ times \ max \ max \ left \ lest \ lest {\ | g（a） ＆+\ max \ left \ {\ | f（a）\ |，\ | f（b）\ | \ right \} \ times \ times \ max \ max \ lest \ left \ {\ | g'（a）\ |，\ |，\ | g'（b） \ end {align*}
Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n f(A)|||\leq\|f^{(n)}(A)\|$ and $\|f^{(n)}(\cdot)\|$ is a quasi-convex function on the set of all positive definite operators in $B(H)$. We establish some estimates of the right hand side of some Hermite-Hadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasi-convex or $s$-convex. As applications, we obtain some of bounds for $|||f(B)-f(A)|||$ in term of $|||B-A|||$. For instance, Let $f,g$ be two operator monotone functions on $(0,\infty)$. Then, for every unitarily invariant norm $|||.|||$ and every positive definite operators $A,B$, \begin{align*} &\left|\left|\left|f(A)g(A)-f(B)g(B)\right|\right|\right|\notag\\ &\leq|||B-A|||\Big[\max\left\{\|f'(A)\|,\|f'(B)\|\right\}\times\max\left\{\|g(A)\|,\|g(B)\|\right\}\notag\\ &+\max\left\{\|f(A)\|,\|f(B)\|\right\}\times \max\left\{\|g'(A)\|,\|g'(B)\|\right\}\Big]. \end{align*}
