Generalized Eigenfunction Expansions for Operator Algebras^)

A generalized eigenfunction expansion may be regarded as a representation for the spectral theorem by a transform technique. These representations have been presented in many forms, an early version of which was the von Neumann "direct integral" decomposition for a class of operator algebras [l9]. In 1953 [17], Mautner applied the von Neumann technique to the class of operators acting in an L2-space whose spectral projections corresponding to the bounded Borel sets are represented by Carleman kernels. These results were also obtained by Bade and Schwartz [l] in 1956 through the use of the Dunford-Pettis theorem [8]. Also, Nelson [l8] in 1958 obtained these results in the form of an operator decomposition by a kernel representation of the operators involved. Using the techniques of Mautner, Browder and Gàrding [3; 9] in 1954 were the first to obtain an eigenfunction expansion for any self-adjoint realization of an elliptic partial differential operator. A somewhat different technique was introduced in 1955 when Gelfand and Kostyucenko obtained a general theorem for the existence of an eigenfunction expansion for a self-adjoint realization of any partial differential operator. Their proof employed a vector-valued differentiation theorem, proved first for Hubert spaces by Birkhoff [2] and for general reflexive Banach spaces by Gelfand [ll]. In 1956, Gelfand and Silov [13] seemed to have been the first to observe the vast generality of these existence theorems and the possibility of a semi-algebraic formulation of the basic hypotheses used. Most striking, perhaps, was the fact that the existence theorem for an eigenfunction expansion could be completely independent of the type of operator considered, and a function only of the topological nature of the subspace over which the expansion takes place. Gelfand and Silov essentially use the nuclearity of the space over which the expansion exists to verify the hypotheses of the GelfandBirkhoff differentiation theorem. All of the existence theorems for generalized eigenfunction expansions mentioned above have been highly analytical. A focal point of our investigation might be regarded as the attempt to replace these analytical techniques by the semi-algebraic techniques suggested by Grothendieck's thesis [14] and Gelfand and Silov [13]. Our results, in essence, prove that in the context