## Advances in Operator Theory

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

### On the numerical radius of a quaternionic normal operator

#### Abstract

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

**Source**

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

**Dates**

Received: 22 November 2016

Accepted: 26 January 2017

First available in Project Euclid: 4 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1512431515

**Digital Object Identifier**

doi:10.22034/aot.1611-1060

**Mathematical Reviews number (MathSciNet)**

MR3730356

**Zentralblatt MATH identifier**

1367.47007

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aot/1512431515