Generalized Cartesian decomposition and numerical radius inequalities

Abstract

Let ${\mathbb {T}} = \{ \lambda \in {\mathbb {C}}: \mid \lambda \mid = 1\}.$ Every linear operator T on a complex Hilbert space ${\mathcal {H}}$ can be decomposed as

$$\begin{aligned} T=\frac{T+\lambda T^*}{2}+ i \frac{T-\lambda T^*}{2i} \,\, \, (\lambda \in {\mathbb {T}}), \end{aligned}$$

designated as the generalized Cartesian decomposition of T. Using the generalized Cartesian decomposition we obtain several lower and upper bounds for the numerical radius of bounded linear operators which refine the existing bounds. We prove that if T is a bounded linear operator on ${\mathcal {H}},$ then

$$\begin{aligned} w(T)\ge & {} \frac{1}{2} \left\| T+ \frac{\lambda +\mu }{2}T^* \right\| , \,\,\text {for all } \lambda ,\,\mu \in {\mathbb {T}}. \end{aligned}$$

This improves the existing bounds $w(T)\ge \frac{1}{2}\Vert T\Vert$, $w(T)\ge \Vert Re(T)\Vert$, $w(T)\ge \Vert Im(T)\Vert$ and so $w^2(T)\ge \frac{1}{4} \Vert T^*T+TT^*\Vert ,$ where Re(T) and Im(T) denote the the real part and the imaginary part of T, respectively. Further, using a lower bound for the numerical radius of a bounded linear operator, we develop upper bounds for the numerical radius of the commutator of operators which generalize and improve on the existing ones.