Representation and normality of hyponormal operators in the closure of \(\mathcal{AN}\)-operators

Abstract

A bounded linear operator \(T\) on a Hilbert space \(H\) is said to be absolutely norm attaining \((T \in \mathcal{AN}(H))\) if the restriction of \(T\) to any non-zero closed subspace attains its norm and absolutely minimum attaining \((T \in \mathcal{AM}(H))\) if every restriction to a non-zero closed subspace attains its minimum modulus.

In this article, we characterize normal operators in \(\overline{\mathcal{AN}(H)}\), the operator norm closure of \(\mathcal{AN}(H)\), in terms of the essential spectrum. Later, we study representations of quasinormal and hyponormal operators in \(\overline{\mathcal{AN}(H)}\). Explicitly, we prove that any hyponormal operator in \(\overline{\mathcal{AN}(H)}\) is a direct sum of a normal \(\mathcal{AN}\)-operator and a \(2\times2\) upper triangular \(\mathcal{AM}\)-operator matrix. Finally, we deduce some sufficient conditions implying the normality of them.