A-Browder-type theorems for direct sums of operators

Berkani, Mohammed, Sarih, Mustapha, and Zariouh, Hassan. "A-Browder-type theorems for direct sums of operators." Mathematica Bohemica 141.1 (2016): 99-108. <http://eudml.org/doc/276812>.

@article{Berkani2016,
abstract = {We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $(\rm SBaw)$, $(\rm SBab)$, $(\rm SBw)$ and $(\rm SBb)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $(\rm SBab)$, then $S\oplus T$ has the property $(\rm SBab)$ if and only if $\sigma _\{\rm SBF_+^-\}(S\oplus T)=\sigma _\{\rm SBF_+^-\}(S)\cup \sigma _\{\rm SBF_+^-\}(T)$, where $\sigma _\{\rm SBF_\{+\}^\{-\}\}(T)$ is the upper semi-B-Weyl spectrum of $T$. We obtain analogous preservation results for the properties $(\rm SBaw)$, $(\rm SBb)$ and $(\rm SBw)$ with extra assumptions.},
author = {Berkani, Mohammed, Sarih, Mustapha, Zariouh, Hassan},
journal = {Mathematica Bohemica},
keywords = {property $(\rm SBaw)$; property $(\rm SBab)$; upper semi-B-Weyl spectrum; direct sum},
language = {eng},
number = {1},
pages = {99-108},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A-Browder-type theorems for direct sums of operators},
url = {http://eudml.org/doc/276812},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Berkani, Mohammed
AU - Sarih, Mustapha
AU - Zariouh, Hassan
TI - A-Browder-type theorems for direct sums of operators
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 1
SP - 99
EP - 108
AB - We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $(\rm SBaw)$, $(\rm SBab)$, $(\rm SBw)$ and $(\rm SBb)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $(\rm SBab)$, then $S\oplus T$ has the property $(\rm SBab)$ if and only if $\sigma _{\rm SBF_+^-}(S\oplus T)=\sigma _{\rm SBF_+^-}(S)\cup \sigma _{\rm SBF_+^-}(T)$, where $\sigma _{\rm SBF_{+}^{-}}(T)$ is the upper semi-B-Weyl spectrum of $T$. We obtain analogous preservation results for the properties $(\rm SBaw)$, $(\rm SBb)$ and $(\rm SBw)$ with extra assumptions.
LA - eng
KW - property $(\rm SBaw)$; property $(\rm SBab)$; upper semi-B-Weyl spectrum; direct sum
UR - http://eudml.org/doc/276812
ER -