On the Generalized Joint Eigenvector Expansion for Commuting Normal Operators

Abstract

Let ${e_\lambda }(x) = {e^{i2\pi \lambda x}},\lambda \in \Lambda ,$ be a family of commuting normal operators in a separable Hilbert space Ho. Obtaining the spectral expansion of A involves the construction of the corresponding joint resolution of identity E. The support, supp E, is not, generally, a set of full E-measure. This causes numerous difficulties, in particular, when proving the projection spectral theorem, i.e., the main theorem about the expansion in generalized joint eigenvectors. In this review, we provide an example of a joint resolution of identity with an empty support and show supp E has a full outer measure under the conditions of the projection spectral theorem. This result can be used to simplify the proof of the theorem and to refine its assertions.