Spectral decomposition of normal absolutely minimum attaining operators

TL;DR

This paper characterizes positive absolutely minimum attaining operators via their essential spectrum, explores conditions for their adjoints, and establishes a spectral decomposition for normal such operators, advancing the understanding of their spectral properties.

Contribution

It provides a new spectral characterization of positive absolutely minimum attaining operators and a spectral decomposition for normal ones, including conditions for their adjoints.

Findings

Characterization of positive absolutely minimum attaining operators via essential spectrum

Spectral decomposition of normal absolutely minimum attaining operators

Conditions under which the adjoint of an dcm-operator is dcm

Abstract

Let $T:H_1\rightarrow H_2$ be a bounded linear operator defined between complex Hilbert spaces $H_1$ and $H_2$. We say $T$ to be \textit{minimum attaining} if there exists a unit vector $x\in H_1$ such that $\|Tx\|=m(T)$, where $m(T):=\inf{\{\|Tx\|:x\in H_1,\; \|x\|=1}\}$ is the \textit{minimum modulus} of $T$. We say $T$ to be \textit{absolutely minimum attaining} ($\mathcal{AM}$-operators in short), if for any closed subspace $M$ of $H_1$ the restriction operator $T|_M:M\rightarrow H_2$ is minimum attaining. In this paper, we give a new characterization of positive absolutely minimum attaining operators ($\mathcal{AM}$-operators, in short), in terms of its essential spectrum. Using this we obtain a sufficient condition under which the adjoint of an $\mathcal{AM}$-operator is $\mathcal{AM}$. We show that a paranormal absolutely minimum attaining operator is hyponormal. Finally, we establish a spectral decomposition of normal absolutely minimum attaining operators. In proving all these results we prove several spectral results for paranormal operators. We illustrate our main result with an example.