Borel structure of the spectrum of a closed operator

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 $λ\in σ_p(T)$ such that $\dim \ker (T - λ) = n$ and let $σ_{p[\infty]}(T)$ be the set of all $λ\in σ_p(T)$ for which $\ker (T - λ)$ is infinite-dimensional. It is shown that $σ_p(T)$ is $\mathcal{F}_σ$, $σ_{p[\infty]}(T)$ is $\mathcal{F}_{σδ}$ and for each finite $n$ the set $σ_{p[n]}(T)$ is the intersection of an $\mathcal{F}_σ$ and a $\mathcal{G}_δ$ 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 $\mathcal{H}$ more detailed decomposition of the spectra is done and the algebra of all bounded linear operators on $\mathcal{H}$ is decomposed into Borel parts. In particular, it is shown that the set of all closed range operators on $\mathcal{H}$ is Borel.

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