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

Piotr Niemiec

Studia Mathematica, Tome 209 (2012), p. 77-85 / Harvested from The Polish Digital Mathematics Library

Résumé

For a linear operator T in a Banach space let $σ_{p}(T)$ denote the point spectrum of T, let $σ_{p,n}(T)$ 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 ker(T - λ) is infinite-dimensional. It is shown that $σ_{p}(T)$ is $ℱ_{σ}$, $σ_{p,∞}(T)$ is $ℱ_{σδ}$ and for each finite n the set $σ_{p,n}(T)$ is the intersection of an $ℱ_{σ}$ set and a $_{δ}$ 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 a more detailed decomposition of the spectra is obtained and the algebra of all bounded linear operators on is decomposed into Borel parts. In particular, it is shown that the set of all closed range operators on is Borel.

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Publié le : 2012-01-01

Zbl 1259.47004

EUDML-ID : urn:eudml:doc:286530