Unbounded quasinormal operators revisited

TL;DR

This paper explores unbounded quasinormal operators, establishing their characterizations, showing equivalences between different definitions, and providing examples to demonstrate the sharpness of these results.

Contribution

It clarifies the definition of unbounded quasinormal operators and proves their equivalence with other existing definitions, enriching the theoretical understanding.

Findings

Characterizations of unbounded quasinormal operators established

Equivalence of Kaufman's and other definitions proven

Examples demonstrating the sharpness of results provided

Abstract

Various characterizations of unbounded closed densely defined operators commuting with the spectral measures of their moduli are established.In particular, Kaufman's definition of an unbounded quasinormal operator is shown to coincide with that given by the third-named author and Szafraniec. Examples demonstrating the sharpness of results are constructed.