摘要:
Let M be a Bade complete (or sigma-complete) Boolean algebra of projections in a Banach space X. This paper is concerned with the following questions: When is M equal to the resolution of the identity (or the strong operator closure of the resolution of the identity) of some scalar-type spectral operator T (with sigma(T) subset of or equal to R) in X? It is shown that if X is separable, then M always coincides with such a resolution of the identity. For certain restrictions on M some positive results are established in non-separable spaces X. An example is given for which M is neither a resolution of the identity nor the strong operator closure of a resolution of the identity.