QUASINORMAL EXTENSIONS OF SUBNORMAL OPERATORS

This paper is dedicated to the memory of John S. MacNerney ABSTRACT. Relations between subnormal, quasinormal and normal operators on Hilbert space are investigated. It is shown that every subnormal operator has a minimal quasinormal extension which may be used to lift operators in the cornmutant of the subnormal to the cornmutant of the minimal normal extension. Necessary and sufficient conditions, implying the quasinormality of a restriction of a normal operator, are developed. Introduction. Quasinormal operators were introduced by A. Brown in [3]. They are exactly the continuous linear operators A on a Hilbert space which commute with A*A. The purpose of this paper is to study certain relations between subnormal, quasinormal and normal operators. The class of subnormal operators has been the subject of extensive investigation since its introduction in 1950 [9]. The less genera