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

• Autorzy: Piotr Niemiec
• Języki publikacji: EN

Abstrakty

EN

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 $𝒢_{δ}$ 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 𝓗 a more detailed decomposition of the spectra is obtained and the algebra of all bounded linear operators on 𝓗 is decomposed into Borel parts. In particular, it is shown that the set of all closed range operators on 𝓗 is Borel.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2012
• Tom: 208
• Numer: 1
• Strony: 77-85

Daty

wydano

2012

Twórcy

autor

Piotr Niemiec

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

Identyfikatory

DOI

10.4064/sm208-1-5