Borel structure of the spectrum of a closed operator

TL;DR

This paper investigates the Borel complexity of the point spectrum of linear operators in Banach and Hilbert spaces, providing detailed spectral decompositions and classifying various spectral sets within the Borel hierarchy.

Contribution

It offers new results on the Borel classification of spectral subsets for closed operators, including detailed decompositions and the Borel nature of the set of closed range operators.

Findings

The point spectrum is an _{\u03c6} set.

The infinite-dimensional point spectrum is _{a4}.

The set of all closed range operators is Borel.

Abstract

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 $\sigma_{p[\infty]}(T)$ 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 $\mathcal{F}_{\sigma}$, $\sigma_{p[\infty]}(T)$ 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.