Representation and normality of Hyponormal operators in the closure of $\mathcal{A}\mathcal{N}$-operators

Let $H_1$, $H_2$ be complex Hilbert spaces. A bounded linear operator $T:H_1\to H_2$ is said to be norm attaining if there exists a unit vector $x\in H_1$ such that $\mathrm{\parallel }Tx\mathrm{\parallel }=\mathrm{\parallel }T\mathrm{\parallel }$. If $T{\mathrm{\mid }}_M:M\to H_2$ is norm attaining for every closed subspace $M$ of $H_1$, then we say that $T$ is an absolutely norm attaining ($\mathcal{A}\mathcal{N}$-operator). If the norm of the operator is replaced by the minimum modulus $m(T)=\inf \{\mathrm{\parallel }Tx\mathrm{\parallel }:x\in H_1,\mathrm{\parallel }x\mathrm{\parallel }=1\}$, then $T$ is said to be a minimum attaining and an absolutely minimum attaining operator ($\mathcal{A}\mathcal{M}$-operator), respectively. In this article, we give representations of quasinormal $\mathcal{A}\mathcal{N}$, $\mathcal{A}\mathcal{M}$-operators and the operators in the closure of these two classes. Later we extend these results to the class of hyponormal operators in the closure of $\mathcal{A}\mathcal{N}$-operators and a further look at some sufficient conditions under which these operators become normal.