Spectral properties of absolutely minimum attaining operators

In this article, we give a new characterization of positive absolutely minimum attaining operators in terms of essential spectrum. We discuss absolutely minimum attaining property of multiplication operator on a σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-finite measure space and then establish a spectral decomposition for normal absolutely minimum attaining operators. We also give a sufficient condition under which the adjoint of an absolutely minimum attaining operator is absolutely minimum attaining and show that a paranormal absolutely minimum attaining operator with an extra condition is hyponormal. In the end, we prove a few spectral properties of paranormal operators.