Zamani, A. (2017). Some lower bounds for the numerical radius of Hilbert space operators. Advances in Operator Theory, 2(2), 98-107. doi: 10.22034/aot.1612-1076
Ali Zamani. "Some lower bounds for the numerical radius of Hilbert space operators". Advances in Operator Theory, 2, 2, 2017, 98-107. doi: 10.22034/aot.1612-1076
Zamani, A. (2017). 'Some lower bounds for the numerical radius of Hilbert space operators', Advances in Operator Theory, 2(2), pp. 98-107. doi: 10.22034/aot.1612-1076
Zamani, A. Some lower bounds for the numerical radius of Hilbert space operators. Advances in Operator Theory, 2017; 2(2): 98-107. doi: 10.22034/aot.1612-1076
Some lower bounds for the numerical radius of Hilbert space operators
We show that if $T$ is a bounded linear operator on a complex Hilbert space, then \begin{equation*} \frac{1}{2}\Vert T\Vert\leq \sqrt{\frac{w^2(T)}{2} + \frac{w(T)}{2}\sqrt{w^2(T) - c^2(T)}} \leq w(T), \end{equation*} 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 $k$, \begin{equation*} \frac{(1 + 2t)^{\frac{1}{2k}}}{{2}^{\frac{1}{k}}}m(T)\leq w(T), \end{equation*} where $m(T)$ denotes the minimum modulus of $T$. Some other related results are also presented.