Advances in Operator Theory

On the numerical radius of a quaternionic normal operator

Golla Ramesh

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We prove that for a right linear bounded normal operator on a quaternionic Hilbert space (quaternionic bounded normal operator) the norm and the numerical radius are equal. As a consequence of this result we give a new proof of the known fact that a non zero quaternionic compact normal operator has a non zero right eigenvalue. Using this we give a new proof of the spectral theorem for quaternionic compact normal operators. Finally, we show that every quaternionic compact operator is norm attaining and prove the Lindenstrauss theorem on norm attaining operators, namely, the set of all norm attaining quaternionic operators is norm dense in the space of all bounded quaternionic operators defined between two quaternionic Hilbert spaces.

Article information

Adv. Oper. Theory, Volume 2, Number 1 (2017), 78-86.

Received: 22 November 2016
Accepted: 26 January 2017
First available in Project Euclid: 4 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47S10: Operator theory over fields other than $R$, $C$ or the quaternions; non- Archimedean operator theory
Secondary: 43B15 35P05: General topics in linear spectral theory

quaternionic Hilbert space normal operator compact operator right eigenvalue norm attaining operator Lindenstrauss theorem


Ramesh, Golla. On the numerical radius of a quaternionic normal operator. Adv. Oper. Theory 2 (2017), no. 1, 78--86. doi:10.22034/aot.1611-1060.

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