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

Abstract

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 $\Vert Tx\Vert =\Vert T\Vert$. If $T{|}_M:M\to H_2$ 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 $m(T)=\inf \{\Vert Tx\Vert :x\in H_1,\Vert x\Vert =1\}$, 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.

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