Spectral decomposition of normal absolutely minimum attaining operators

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.