An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality

TL;DR

This paper derives new sharp inequalities relating the numerical radius and operator norms of Hilbert space operators, extending previous results through convexity and functional analysis techniques.

Contribution

It introduces generalized numerical radius inequalities involving operator powers, convex functions, and sums, improving bounds for Hilbert space operators.

Findings

Established new bounds for the numerical radius involving operator norms.

Extended classical inequalities using convexity and functional calculus.

Provided conditions under which the inequalities are sharp or optimal.

Abstract

We provide a number of sharp inequalities involving the usual operator norms of Hilbert space operators and powers of the numerical radii. Based on the traditional convexity inequalities for nonnegative real numbers and some generalize earlier numerical radius inequalities, operator. Precisely, we prove that if $\A_i,\B_i,\X_i\in\bh$ ($i=1,2,\cdots,n$), $m\in\N$, $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ and $\phi$ and $\psi$ are non-negative functions on $[0,\infty)$ which are continuous such that $\phi(t)\psi(t)=t$ for all $t \in [0,\infty)$, then \begin{equation*} w^{2r}\bra{\sum_{i=1}^{n}\X_i\A_i^m\B_i}\leq \frac{n^{2r-1}}{m}\sum_{j=1}^{m}\norm{\sum_{i=1}^{n}\frac{1}{p}S_{i,j}^{pr}+\frac{1}{q}T_{i,j}^{qr}}-r_0\inf_{\norm{x}=1}\rho(\xi), \end{equation*} where $r_0=\min\{\frac{1}{p},\frac{1}{q}\}$, $S_{i,j}=\X_i\phi^2\bra{\abs{\A_i^{j*}}}\X_i^*$, $T_{i,j}=\bra{\A_i^{m-j}\B_i}^*\psi^2\bra{\abs{\A_i^j}}\A_i^{m-j}\B_i$ and $$\rho(x)=\frac{n^{2r-1}}{m}\sum_{j=1}^{m}\sum_{i=1}^{n}\bra{\seq{S_{i,j}^r\xi,\xi}^{\frac{p}{2}}-\seq{T_{i,j}^r\xi,\xi}^{\frac{q}{2}}}^2.$$