SUPPORT OF A JOINT RESOLUTION OF IDENTITY AND THE PROJECTION SPECTRAL THEOREM

Abstract

Let A = (Ax)x ∈ Xbe a family of commuting normal operators in a separable Hilbert space H0. Obtaining the spectral expansion of A involves constructing of the corresponding joint resolution of identity E. The support supp E is not, in general, a set of full 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 work, we show that supp E has a full outer measure under the conditions of the projection spectral theorem. Using this result, we simplify the proof of the theorem and refine its assertions.