Quasinormal and hyponormal operators in the β(H)-class

@misc{golla2026quasinormal,
  title = {{Quasinormal and hyponormal operators in the $β(H)$-class}},
  author = {Ramesh Golla and Hiroyuki Osaka and Shanola S. Sequeira},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13490},
  eprint = {2610.13490},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13490},
}

TY  - UNPB
TI  - Quasinormal and hyponormal operators in the β(H)-class
AU  - Ramesh Golla
AU  - Hiroyuki Osaka
AU  - Shanola S. Sequeira
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13490
UR  - https://arxiv.org/abs/2610.13490
AB  - Following \cite{BR} we study hyponormal operators in the strictly larger class β(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=⨁_iλ_iS_i (with isometries S_i) for the β(H)-operators whose |T|-eigenspaces are reducing, equivalently the quasinormal operators with |T|∈β(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 β(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.
ER  -