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

TL;DR

This paper develops new norm and numerical radius inequalities for sums of Hilbert space operators, extending recent results and providing bounds for normal operators using functional calculus.

Contribution

It introduces generalized inequalities for operator sums and numerical radius, broadening the scope of existing bounds with new functional relationships.

Findings

Established a new upper bound for the norm of the sum of normal operators.

Derived inequalities for the numerical radius involving operator functions.

Generalized previous results to broader classes of operators and functions.

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.