On Normal Extensions of Unbounded Operators. III. Spectral Properties

Positive definiteness and commutativity of operators

“…However if A has sufficiently many analytic or quasianalytic vectors, then the converse implication holds (cf. [16], [9]); in particular this is the case for A ∈ B(D) n (cf. [8]).…”

“…Since N is formally normal, it must be Q(N ) = Q(N # ). Hence, by [16,Theorem 1], both the operators N and N # are normal. Consequently N * is normal.…”

“…The Ito theorem says that such a family is subnormal if and only if it is positive definite in the sense of Halmos (H-positive definite for short). This result has further been generalized to the case of families of unbounded operators by the author and Szafraniec [16], and independently by Jorgensen [9]. Contrary to [16], the families of operators taken into consideration in [9] have not been assumed to be commutative; commutativity follows from H-positive definiteness due to [9,Theorem 4.2]:…”

On a Friedrichs Extension Related to Unbounded Subnormal Operators

“…Our methods make an essential use of the fact that the multiplication operators under consideration are unbounded cyclic subnormal operators (see definitions below), and rely crucially on the theory of such operators as expounded in [9]. (See also [7], [8].) The interaction of the theory of sectorial sesquilinear forms with that of unbounded subnormals was explored by the authors in [2] with a special attention paid to those subnormal operators that admit 'analytic models'.…”

“…The present paper is an illustration of the utility of that theory in the study of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Our methods make an essential use of the fact that the multiplication operators under consideration are unbounded cyclic subnormal operators (see definitions below), and rely crucially on the theory of such operators as expounded in [9]. (See also [7], [8].)…”