Spectral decomposition of normal absolutely minimum attaining operators

Abstract

Let $T:H_1\to H_2$ 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 $\Vert Tx\Vert =m(T)$, where $m(T):=\inf \{\Vert Tx\Vert :x\in H_1,\Vert x\Vert =1\}$ is the \textit{minimum modulus} of $T$. We say $T$ to be \textit{absolutely minimum attaining} ($\mathcal{A}\mathcal{M}$-operators in short), if for any closed subspace $M$ of $H_1$ the restriction operator $T{|}_M:M\to H_2$ is minimum attaining. In this paper, we give a new characterization of positive absolutely minimum attaining operators ($\mathcal{A}\mathcal{M}$-operators, in short), in terms of its essential spectrum. Using this we obtain a sufficient condition under which the adjoint of an $\mathcal{A}\mathcal{M}$-operator is $\mathcal{A}\mathcal{M}$. 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.

The hypothesis in Theorem 4.7 is changed and hence the title of the article