Quasinormal and hyponormal operators in the $\beta (H)$-class

Following \cite{BR} we study hyponormal operators in the strictly larger class $\beta (H)$ of operators whose restriction to every non-zero reducing subspace is norm attaining. The Weyl-type characterization of positive operators via the essential spectrum used in the absolutely norm attaining case is replaced by a well-ordering condition on the eigenvalues (Theorem~\ref{thm:positive}). We prove a transfinite, block-diagonal representation $T={⨁}_i{\lambda }_i{S}_i$ (with isometries $S_i$) for the $\beta (H)$-operators whose $\mathrm{\mid }T\mathrm{\mid }$-eigenspaces are reducing, equivalently the quasinormal operators with $\mathrm{\mid }T\mathrm{\mid }\in \beta (H)$, and we show by examples that hyponormality alone does not give such a decomposition. For normal operators we obtain a complete representation, and for hyponormal $\beta (H)$-operators we show that normality is equivalent to the spectrum having planar Lebesgue measure zero. An analogous area-zero criterion for normality is proved for hyponormal operators attaining the minimum modulus on every non-zero reducing subspace.