Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec¹

¹ Instytut Matematyki Wydział Matematyki i Informatyki Uniwersytet Jagielloński Łojasiewicza 6 30-348 Kraków, Poland

Studia Mathematica, Tome 208 (2012) no. 1, pp. 77-85

Résumé

For a linear operator $T$ in a Banach space let $\sigma_p(T)$ denote the point spectrum of $T$, let $\sigma_{p,n}(T)$ 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 $\mathcal{F}_{\sigma}$ set 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 $\mathcal{H}$ a more detailed decomposition of the spectra is obtained 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.

Détail

DOI : 10.4064/sm208-1-5

Keywords: linear operator banach space sigma denote point spectrum nbsp sigma finite set lambda sigma dim ker lambda sigma infty set lambda sigma which ker lambda infinite dimensional shown sigma mathcal sigma sigma infty mathcal sigma delta each finite set sigma intersection mathcal sigma set mathcal delta set provided closable domain separable weakly sigma compact closed densely defined operators separable hilbert space mathcal detailed decomposition spectra obtained algebra bounded linear operators mathcal decomposed borel parts particular shown set closed range operators mathcal borel