Borel structure of the spectrum of a closed operator

Abstract

For a linear operator T in a Banach space let σ p (T ) denote the point spectrum of T , σ p[n] (T ) for finite n > 0 be the set of all λ ∈ σ p (T ) such that dim ker(T − λ) = n and let σ p[∞] (T ) be the set of all λ ∈ σ p (T ) for which keris the intersection of an F σ and a G δ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space H more detailed decomposition of the spectra is done and the algebra of all bounded linear operators on H is decomposed into Borel parts. In particular, it is shown that the set of all closed range operators on H is Borel.