Borel structure of the spectrum of a closed operator

For a linear operator $T$ in a Banach space let ${\sigma }_p(T)$ denote the point spectrum of $T$, ${\sigma }_{p[n]}(T)$ for finite $n>0$ be the set of all $\lambda \in {\sigma }_p(T)$ such that $\dim \ker (T-\lambda )=n$ and let ${\sigma }_{p[\mathrm{\infty }]}(T)$ be the set of all $\lambda \in {\sigma }_p(T)$ for which $\ker (T-\lambda )$ is infinite-dimensional. It is shown that ${\sigma }_p(T)$ is ${\mathcal{F}}_{\sigma }$, ${\sigma }_{p[\mathrm{\infty }]}(T)$ is ${\mathcal{F}}_{\sigma \delta }$ and for each finite $n$ the set ${\sigma }_{p[n]}(T)$ is the intersection of an ${\mathcal{F}}_{\sigma }$ and a ${\mathcal{G}}_{\delta }$ set provided $T$ is closable and the domain of $T$ is separable and weakly $\sigma$-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.