Further norm and numerical radius inequalities for sum of Hilbert space operators

Davood Afraz, Ramatollah Lashkaripour, Mojtaba Bakherad


Let ${\mathbb B}(\mathcal H)$ denote the set of all bounded linear operators on a complex Hilbert space ${\mathcal H}$. In this paper, the authors present some norm inequalities for sum of operators which are a generalization of some recent results. Among other inequalities, it is shown that if $S, T\in {\mathbb B}({\mathcal H})$ are normal operators, then
\left\Vert S+T\right\Vert&\leq& \frac{1}{2}(\left\Vert S\right\Vert+\left\Vert T\right\Vert)+\frac{1}{2}\sqrt{ (\left\Vert S \right\Vert-\left\Vert T\right\Vert)^2+
4\left\Vert f_1(\vert S \vert)g_1(\vert T\vert)\right\Vert \left\Vert f_2(\vert S \vert)g_2(\vert T\vert) \right\Vert},
where $f_1,f_2,g_1,g_2$ are non-negative continuous functions on $[0,\infty )$, in which $f_1(x)f_2(x)=x$ and $g_1(x)g_2(x)=x\,(x\geq 0)$. Moreover, several inequalities for the numerical radius are shown.


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