On the numerical radius of a quaternionic normal operator

TL;DR

This paper establishes that for quaternionic normal operators, the norm equals the numerical radius, and explores implications for spectral theory, eigenvalues, and norm attainment in quaternionic Hilbert spaces.

Contribution

It proves the equality of norm and numerical radius for quaternionic normal operators and provides new proofs for spectral properties and norm attainment results.

Findings

Norm equals numerical radius for quaternionic normal operators

Non-zero quaternionic compact normal operators have non-zero eigenvalues

Quaternionic compact operators are norm attaining and dense in the space of bounded operators

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.