Spectral Operators in Spaces of Distributions

The purposes of this note are to complement the results of [4] by citing examples of spectral operators (in the sense of that paper) on non-normable locally convex spaces and to announce a result on the structure of spectral operators which indicates that these examples exhibit rather typical behavior for spectral operators on a large class of locally convex spaces of interest to analysts. We retain the notions of [4] and refer to it for proofs. Let E be a locally convex space (always assumed Hausdorff), £(E) the algebra of continuous linear transformations of E into itself (the identity being denoted by e), £S(E) the algebra £(E) under the topology of pointwise convergence, and let £ if R(-) is holomorphic in a neighborhood of oo. The spectrum of u (denoted by of (&(X) into £8(E); conversely, any continuous homomorphism of 6(X) into £a(E) has associated with it a spectral measure jut (and thus an extension to ($>(X)). The connection between the measure and the homomorphism is given by