A Class of Norm Inequalities for Operator Monotone Functions and Hyponormal Operators
Tóm tắt
Let
$$\Psi $$
,
$$\Phi $$
be s.n. functions,
$$p\ge 2,$$
and let
$$\varphi $$
be an operator monotone function on
$$[0,\infty )$$
such that
$$\varphi (0)=0.$$
If
are such that A and B are strictly accretive and
then also
and
$$\begin{aligned}{} & {} \vert {\;\!\!\vert {AX\varphi {(B)}-\varphi {(A)}XB}\vert \;\!\!}\vert _\Psi \\{} & {} \qquad \le \left\| \,\!\sqrt{\varphi \Bigl ({\tfrac{A+A^*}{2}}\Bigr )-\tfrac{A+A^*}{2}\varphi ^\prime \Bigl ({\tfrac{A+A^*}{2}}\Bigr )} \Bigl ({\tfrac{A+A^*}{2}}\Bigr )^{-1}\!A(AX-XB)B\Bigl ({\tfrac{B+B^*}{2}}\Bigr )^{-1}\!\!\right. \\{} & {} \qquad \quad \left. \sqrt{\varphi \Bigl ({\tfrac{B+B^*}{2}}\Bigr )-\tfrac{B+B^*}{2}\varphi ^\prime \Bigl ({\tfrac{B+B^*}{2}}\Bigr )}\,\right\| _\Psi \,\!\!. \end{aligned}$$
under any of the following conditions:
Alternative inequalities for
$$\vert {\;\!\!\vert {\cdot }\vert \;\!\!}\vert _{\Phi ^{(p)}}$$
norms are also obtained.
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