Some lower bounds for the numerical radius of Hilbert space operators

Abstract

We show that if $T$ is a bounded linear operator on a complex Hilbert space, then $$\frac{1}{2} ||T|| \leq \sqrt {{\frac{w^2(T)}{2}} + \frac{w(T)}{2} \sqrt{w^2(T) - c^2(T)}} \leq w(T),$$ where $w(\cdot)$ and $c(\cdot)$ are the numerical radius and the Crawford number, respectively. We then apply it to prove that for each $t \in [0, \frac {1}{2})$ and natural number $k$, $$\frac {(1 + 2t)^{\frac{1}{2k}}}{{2}^{\frac{1}{k}}}m(T)\leq w(T),$$ where $m(T)$ denotes the minimum modulus of $T$. Some other related results are also presented.