Advances in Operator Theory

Some lower bounds for the numerical radius of Hilbert space operators

Ali Zamani

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aot/1512431558

Digital Object Identifier
doi:10.22034/aot.1612-1076

Mathematical Reviews number (MathSciNet)
MR3730061

Zentralblatt MATH identifier
1367.47009

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

Keywords
numerical radius operator norm inequality Cartesian decomposition

Citation

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. https://projecteuclid.org/euclid.aot/1512431558


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