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

Piotr Niemiec. "Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space." Studia Mathematica 208.1 (2012): 77-85. <http://eudml.org/doc/286530>.

@article{PiotrNiemiec2012,
abstract = {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.},
author = {Piotr Niemiec},
journal = {Studia Mathematica},
keywords = {spectrum; point spectrum; Hilbert space; reflexive Banach space; Borel set; closable operator; weak topology},
language = {eng},
number = {1},
pages = {77-85},
title = {Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space},
url = {http://eudml.org/doc/286530},
volume = {208},
year = {2012},
}

TY - JOUR
AU - Piotr Niemiec
TI - Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space
JO - Studia Mathematica
PY - 2012
VL - 208
IS - 1
SP - 77
EP - 85
AB - 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.
LA - eng
KW - spectrum; point spectrum; Hilbert space; reflexive Banach space; Borel set; closable operator; weak topology
UR - http://eudml.org/doc/286530
ER -