Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas ( IF 1.8 ) Pub Date : 2021-04-02 , DOI: 10.1007/s13398-021-01037-z Silvestru Sever Dragomir
For a continuous and positive function \(w\left( \lambda \right) ,\) \(\lambda >0\) and \(\mu \) a positive measure on \((0,\infty )\) we consider the following convex integral transform
$$\begin{aligned} \mathcal {C}\left( w,\mu \right) \left( T\right) :=\int _{0}^{\infty }w\left( \lambda \right) T^{2}\left( \lambda +T\right) ^{-1}d\mu \left( \lambda \right) , \end{aligned}$$where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among other that, for all A, \(B>0\) with \(BA+AB\ge 0,\)
$$\begin{aligned} \mathcal {C}(w,\mu )\left( A+B\right) \ge \mathcal {C}(w,\mu )\left( A\right) +\mathcal {C}(w,\mu )\left( B\right) . \end{aligned}$$In particular, we have for \(r\in (0,1],\) the power inequality
$$\begin{aligned} \left( A+B\right) ^{r+1}\ge A^{r+1}+B^{r+1} \end{aligned}$$and the logarithmic inequality
$$\begin{aligned} \left( A+B\right) \ln \left( A+B\right) \ge A\ln A+B\ln B. \end{aligned}$$Some examples for operator monotone and operator convex functions and integral transforms \(\mathcal {C}\left( \cdot ,\cdot \right) \) related to the exponential and logarithmic functions are also provided.
中文翻译：
Hilbert空间中正算子的凸积分变换的超加和性
对于连续和正函数\（w \ left（\ lambda \ right），\） \（\ lambda> 0 \）和\（\ mu \）对\（（0，\ infty）\）的正度量，我们考虑以下凸 积分变换
$$ \ begin {aligned} \ mathcal {C} \ left（w，\ mu \ right）\ left（T \ right）：= \ int _ {0} ^ {\ infty} w \ left（\ lambda \ right ）T ^ {2} \ left（\ lambda + T \ right）^ {-1} d \ mu \ left（\ lambda \ right），\ end {aligned} $$假设对于T而言存在积分，则该函数在复Hilbert空间H上为一个正算子。我们证明，对于所有A，\（B> 0 \）与\（BA + AB \ ge 0，\）
$$ \ begin {aligned} \ mathcal {C}（w，\ mu）\ left（A + B \ right）\ ge \ mathcal {C}（w，\ mu）\ left（A \ right）+ \ mathcal {C}（w，\ mu）\ left（B \ right）。\ end {aligned} $$特别地，对于\（r \ in（0,1]，\），我们具有幂不等式
$$ \ begin {aligned} \ left（A + B \ right）^ {r + 1} \ ge A ^ {r + 1} + B ^ {r + 1} \ end {aligned} $$和对数不等式
$$ \ begin {aligned} \ left（A + B \ right）\ ln \ left（A + B \ right）\ ge A \ ln A + B \ ln B. \ end {aligned} $$还提供了一些与指数和对数函数有关的算子单调和算子凸函数以及积分变换\（\ mathcal {C} \ left（\ cdot，\ cdot \ right）\）的示例。
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