Positivity ( IF 0.8 ) Pub Date : 2020-06-16 , DOI: 10.1007/s11117-020-00770-w Amel Bourahli , Omar Hirzallah , Fuad Kittaneh
Let \(T_{11},T_{12},T_{21},\) and \(T_{22}\) be \(n\times n\) complex matrices, \(\ T=\left( \begin{array}{cc} T_{11} &{} T_{12} \\ T_{21} &{} T_{22} \end{array} \right) \) be accretive-dissipative, \(\gamma \in (0,1]\), \(r\ge 2,\ \)and let \( \alpha ,\beta \in [0,1]\) such that \(\alpha +\beta =1\). If f is an increasing convex function on \([0,\infty )\) such that \(f(0)=0\), then
$$\begin{aligned} \left| \left| \left| f\left( \left| T_{12}+\left( 2\alpha -1\right) T_{21}^{*}\right| ^{r}\right) +f\left( 2^{r}\alpha ^{r/2}\beta ^{r/2}\left| T_{21}^{*}\right| ^{r}\right) \right| \right| \right| \le \gamma \left| \left| \left| f\left( \frac{\left| T\right| ^{r}}{\gamma ^{r/2}}\right) \right| \right| \right| \end{aligned}$$for every unitarily invariant norm. In addition, if T is contraction and f is submultiplicative, then
$$\begin{aligned}&\left| \left| \left| \left( f\left( \left| T_{12}+\left( 2\alpha -1\right) T_{21}^{*}\right| ^{2}\right) +f\left( 4\alpha \beta \left| T_{21}^{*}\right| ^{2}\right) \right) \right| \right| \right| \\&\quad \le f^{2}\left( \sqrt{2}\right) \left| \left| \left| f^{r}\left( \left| T_{11}\right| ^{1/2}\right) \right| \right| \right| ^{1/r}\left| \left| \left| f^{s}\left( \left| T_{22}\right| ^{1/2}\right) \right| \right| \right| ^{1/s} \end{aligned}$$for every unitarily invariant norm and for every positive real numbers r, s with \(\frac{1}{r}+\frac{1}{s}=1\). Related inequalities for concave functions are also given.
中文翻译：
耗散性$$ 2 \乘以2 $$ 2×2块矩阵的函数的不变范数不等式
令\（T_ {11}，T_ {12}，T_ {21}，\）和\（T_ {22} \）为\（n乘以n \）复数矩阵，\（\ T = \ left（\开始{array} {cc} T_ {11}＆{} T_ {12} \\ T_ {21}＆{} T_ {22} \ end {array} \ right）\）会产生耗散性，\（\ gamma \ in（0,1] \），\（r \ ge 2，\ \）并令\（\ alpha，\ beta \ in [0,1] \）使得\（\ alpha + \ beta = 1 \ ）。如果f是\（[0，\ infty）\）上的递增凸函数，使得\（f（0）= 0 \），则
$$ \ begin {aligned} \ left | \ left | \ left | f \ left（\ left | T_ {12} + \ left（2 \ alpha -1 \ right）T_ {21} ^ {*} \ right | ^ {r} \ right）+ f \ left（2 ^ {r } \ alpha ^ {r / 2} \ beta ^ {r / 2} \ left | T_ {21} ^ {*} \ right | ^ {r} \ right）\ right | \ right | \ right | \ le \ gamma \ left | \ left | \ left | f \ left（\ frac {\ left | T \ right | ^ {r}} {\ gamma ^ {r / 2}} \ right）\ right | \ right | \ right | \ end {aligned} $$每个统一不变的规范。另外，如果T是收缩且f是可乘的，则
$$ \ begin {aligned}＆\ left | \ left | \ left | \ left（f \ left（\ left | T_ {12} + \ left（2 \ alpha -1 \ right）T_ {21} ^ {*} \ right | ^ {2} \ right）+ f \ left（4 \ alpha \ beta \ left | T_ {21} ^ {*} \ right | ^ {2} \ right）\ right）\ right | \ right | \ right | \\＆\ quad \ le f ^ {2} \ left（\ sqrt {2} \ right）\ left | \ left | \ left | f ^ {r} \ left（\ left | T_ {11} \ right | ^ {1/2} \ right）\ right | \ right | \ right | ^ {1 / r} \左| \ left | \ left | f ^ {s} \ left（\ left | T_ {22} \ right | ^ {1/2} \ right）\ right | \ right | \ right | ^ {1 / s} \ end {aligned} $$对于每个单位不变范数和每个正实数r， s与\（\ frac {1} {r} + \ frac {1} {s} = 1 \）。还给出了凹函数的相关不等式。
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