Open Access
Spring 2017 Some lower bounds for the numerical radius of Hilbert space operators
Ali Zamani
Adv. Oper. Theory 2(2): 98-107 (Spring 2017). DOI: 10.22034/aot.1612-1076


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.


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Ali Zamani. "Some lower bounds for the numerical radius of Hilbert space operators." Adv. Oper. Theory 2 (2) 98 - 107, Spring 2017.


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

zbMATH: 1367.47009
MathSciNet: MR3730061
Digital Object Identifier: 10.22034/aot.1612-1076

Primary: 47A12
Secondary: 47A30

Keywords: Cartesian decomposition , inequality , numerical radius , operator norm

Rights: Copyright © 2017 Tusi Mathematical Research Group

Vol.2 • No. 2 • Spring 2017
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