Advances in Operator Theory

Some lower bounds for the numerical radius of Hilbert space operators

Ali Zamani

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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.

Article information

Adv. Oper. Theory, Volume 2, Number 2 (2017), 98-107.

Received: 9 December 2016
Accepted: 30 January 2017
First available in Project Euclid: 4 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A12: Numerical range, numerical radius
Secondary: 47A30: Norms (inequalities, more than one norm, etc.)

numerical radius operator norm inequality Cartesian decomposition


Zamani, Ali. Some lower bounds for the numerical radius of Hilbert space operators. Adv. Oper. Theory 2 (2017), no. 2, 98--107. doi:10.22034/aot.1612-1076.

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