Spectral decomposition of quasi-Montel spaces

Abstract

In [8] the author showed that Montel spaces have the property that all regular Borel spectral measures with values in their continuous-linear-transformation algebras are necessarily purely atomic. The purpose of this note is to make the observation that by virtue of a theorem of Bartle, Dunford and Schwartz [1] and Grothendieck [3], this property is shared by a significantly larger class of locally convex spaces, namely the quasi-Montel spaces of K. Kera [5]. This class includes the classical Banach space 1 and its subspaces and the gestufte Raiume of Kothe and their subspaces [6]. The result given in this note also has consequences, which we shall mention briefly, in the study of the singular operators of Kantorovitz [4]. Let E [Z] be a locally convex topological vector space, which will be assumed to be boundedly complete; recall that a spectral measure triple in S(E) is a triple (X, 25, ,u) where X is a set, 25 is a a-algebra of subsets of X, and ,u: -->.C(E) is an S(E)-valued set function which is countably additive in the weak operator topology and for which ,u(X) = 1 (the identity transformation) and for any 5, E-C2, Mu(6'e) =-(8),u(E). (X, (25, ,u) is said to be equicontinuous if the values of ,u on (25 are, and to be Baire or Borel if X is a compact Hausdorff space and (5 = e or Q8, its a-algebras of Baire or Borel sets respectively. A Borel spectral measure triple (X, 03, ,u) is said to be regular if (Mu(.)x, x') is a regular Borel measure for each xCE and x'CE' (cf. [8, Proposition 3.18]). A point atom of a Borel measure is a point t CX with .( {I }) #0. For x CE, the cyclic subspace and real cyclic subspace generated by x, denoted by 91Q(x) and J)R(X) respectively, are the smallest closed subspace and closed real subspace of E respectively which contain { u(8)xl}es. [8, Proposition 3.15] shows that both 9NR(X) and 9(x) are complete locally convex spaces when E is boundedly complete in Z, [8, Proposition 3.13 et seq.] that )R(X) is a complete vector lattice when ordered by taking its positive cone to be the closed convex cone generated by { u(8)x1}sE, and also that 91Q(X) = =9RR(X) ED i fR(X) X It is easy to see that only small modifications of the proof of [8, Theorem 4.1] suffice to yield a proof of the slightly stronger-looking