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

Abstract

Let ${\mathbb B}(\mathscr H)$ denote the set of all bounded linear operators on a complex Hilbert space ${\mathscr H}$. In this paper, we present some norm inequalities for sums of operators which are a generalization of some recent results. Among other inequalities, it is shown that if $S, T\in {\mathbb B}({\mathscr H})$ are normal operators, then \begin{eqnarray*} \left\Vert S+T\right\Vert \leq \frac{1}{2}(\left\Vert S\right\Vert+\left\Vert T\right\Vert)+\frac{1}{2}\min_{t>0}\sqrt{ (\left\Vert S \right\Vert-\left\Vert T\right\Vert)^2+ \left\Vert \frac{1}{t} f_1(\vert S \vert)g_1(\vert T\vert)+tf_2(\vert S \vert)g_2(\vert T\vert) \right\Vert^2}, \end{eqnarray*} 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, it is shown several inequalities for the numerical radius.