Applications of the Cauchy-Schwarz inequality for the numerical radius

Résumé

The main goal of this article is to establish several new norm and numerical radius inequalities for operators based on the angle between two vectors in Hilbert space. These enhancements and extensions are achieved through the use of the polar and Cartesian decompositions of operators. In particular, it is proved that, if $X\in \mathscr B\left( \mathscr{H} \right)$ has the polar decomposition $X=U\vert X\vert $ and $\mu(\psi)=\frac{1}{4}(2+\cos\psi \cot\psi \log(\frac{1+\sin\psi}{1-\sin\psi}))$, then
\begin{equation*}
\omega^{2r}(X)\le \mu^{2r}(\theta)\left\Vert \frac{1}{p}f^{2pr}(\vert X \vert)+\frac{1}{q} g^{2qr}(\vert X^*\vert) \right\Vert,
\end{equation*}
where $\theta_{X,x}=\angle_{ f(\vert X\vert) x, g(\vert X\vert)U^* x }$, either $0\le \theta< \theta_{X,x} \le\frac{\pi}{2}$ or $\frac{\pi}{2}\le \theta_{X,x} <\theta\le\pi $ for all unit vectors $x\in \mathscr{H}$, $f,g$ are nonnegative continuous functions on $[0,+\infty) $ satisfying the relation $f(t)g(t)=t \,\,(t \in [0,+\infty))$, $r\ge1$, $p,q>1$ and $\frac{1}{p}+\frac{1}{q}=1$.