Characterization of Absolutely Norm Attaining Compact Hyponormal Operators

Abstract

In this paper, we characterize absolute norm-attainability for compact hyponormal operators. We give necessary and sufficient conditions for a bounded linear compact hyponormal operator on an infinite dimensional complex Hilbert space to be absolutely norm attaining. Moreover, we discuss the structure of compact hyponormal operators when they are self-adjoint and normal. Lastly, we discuss in general, other properties of compact hyponormal operators when they are absolutely norm attaining and their commutators.

Keywords

• Norm-attainability,Compactness,Hyponormal operator

References

1. X. Carvajal and W. Neves, \emph{Operators that achieve the norm}, I. Eq. Oper. Theory, \textbf{72}(2)(2012), 179--195.
2. P. Enflo, J. Kover and L. Smithies, \emph{Denseness for norm attaining operator-valued functions}, Linear Algebra Appl., \textbf{338}(2001), 139--144.
3. P. R. Halmos, \emph{A Hilbert space problem book}, Springer-Verlag, New York, 1982.
4. N. B. Okelo, \emph{The norm attainability of some elementary operators}, Appl. Math. E-Notes, \textbf{13}(2013), 1--7.
5. G. Ramesh, \emph{Absolutely Norm attaining Paranormal operators}, J. Math. Anal. Applic., \textbf{465}(1)(2018), 547--556.