We prove the spectral theorem for locally normal operators in terms of a locally spectral measure. In order to do this, we first obtain some characterisations of local projections and we single out and investigate the concept of a locally spectral measure.
This is an investigation of some basic properties of strictly inductive limits of Hilbert spaces, called locally Hilbert spaces, with respect to their topological properties, the geometry of their subspaces, linear functionals and dual spaces.
We point out a relation between the Arveson's Radon-Nikodym derivative and known similarity results for completely bounded maps. We also consider Jordan type decompositions coming out from Wittstock's Decomposition Theorem and illustrate, by an example, the nonuniqueness of these decompositions.
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