Unbounded normal operators on Banach spaces

Abstract

Introduction. In this paper a theory of unbounded normal operators on complex Banach spaces is developed which generalizes many features of the classical theory of maximal normal operators on Hilbert spaces. In particular the operators studied are closely related to the unbounded scalar type spectral operators introduced by Bade [1] and to a generalization of these operators defined here. The definition of these normal operators is given in terms of generators of strongly continuous groups of isometries contained in a commutative V* algebra [6]. Many of the techniques used were suggested by and generalize results in [21, 5, and 19]. In this paper all Banach spaces (X, '3 etc.) are defined over the complex numbers C, and all operators are (not necessarily bounded, closed, or densely defined) linear transformations with domain and range contained in the same Banach space. An operator R is called self conjugate iff iR generates a strongly continuous group of isometries (cf. [19]). This definition generalizes both the notion of an unbounded self adjoint operator on Hilbert space and a concept introduced by I. Vidav [21] for elements in an arbitrary Banach algebra. A norm closed commutative subalgebra % of the algebra [N] of all everywhere defined bounded operators on X is called a commutative V* algebra if every element S E W can be written as R + iJ with R and J self conjugate elements of W [6]. In ?2, it is shown that the weak closure of such an algebra is again a commutative V* algebra. Also if X is weakly complete a weakly closed commutative V* algebra contains a self conjugate resolution of the identity for each of its elements. A commutative V* algebra with the same property is also constructed when =ID*. The numerical range of an operator on a Banach space is defined in a way resembling that of [17] but more easily applied to unbounded operators. Several criteria are given in ?3 for a densely defined operator with real numerical range to have a self conjugate closure. Self conjugate operators R and J are defined to commute iff the groups generated by iR and iJ commute in the ordinary sense (cf. [20, p. 647]). It is shown that this definition agrees with other ideas of