Spectral decomposition of normal absolutely minimum attaining operators

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 $\mathrm{\parallel }Tx\mathrm{\parallel }=m(T)$, where $m(T):=\inf \{\mathrm{\parallel }Tx\mathrm{\parallel }:x\in H_1,\mathrm{\parallel }x\mathrm{\parallel }=1\}$ is the \textit{minimum modulus} of $T$. We say $T$ to be $\mathit{\text{absolutely}}\mathit{\text{minimum}}\mathit{\text{attaining}}$ ($\mathcal{A}\mathcal{M}$-operators in short), if for any closed subspace $M$ of $H_1$ the restriction operator $T{\mathrm{\mid }}_M:M\to H_2$ 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.