  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.
