Semigroups of Scalar Type Operators on Banach Spaces

Abstract

ABSTRACT. The main result is that if {T(t): t > 0} is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space X and if the resolutions of the identity for T(t) are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure K( ■ ) such that T(t) = { exp (Kt)dK(K), for t > 0. This is a direct generalization of the well-known theorem of Sz. 1. Introduction. In this paper we study strongly continuous semigroups {71(f): t > 0} of scalar type operators on a Banach space and extend some wellknown results about semigroups of normal or selfadjoint operators on a Hilbert space. The main result (Theorem 5.3) generalizes Sz.-Nagy's theorem about semigroups of normal operator [12, Theorem 22.4.2] or [19, §XI 3]. We do not assume that the resolutions of the identity for the operators T(t) generate a bounded Boolean algebra of projections; this is obtained as a result. If one assumes this, a much simpler proof of the main theorem could be given. However, this would be quite restrictive since in general the resolutions of the identity for two commuting scalar type operators do not always generate a bounded Boolean algebra of projections, even on a reflexive space (see [16]).In §6 we extend a theorem of Foias, [11] on semigroups of scalar type operators on a Hilbert space. In §7 we study the continuity of f(T(t)), as a