Affiliation:
1. School of Mathematics and Statistics , 47833 Beijing Institute of Technology , Beijing 102401 , P. R. China
Abstract
Abstract
Let
𝒜
{\mathcal{A}}
be a complex unital Banach algebra and let
R
⊆
𝒜
{R\subseteq\mathcal{A}}
be a non-empty set. This paper defines the property such that R is closed for idempotent decomposition (in short, (CID) property) to explore the spectral decomposition relation. Further, for an upper semiregularity R with (CID) property,
R
D
{R^{D}}
is constructed as an extension of R to axiomatically study the accumulation of
σ
R
(
a
)
{\sigma_{R}(a)}
for any
a
∈
𝒜
{a\in\mathcal{A}}
. At last, several illustrative examples on Banach algebra and operator algebra are provided.
Funder
National Natural Science Foundation of China
Reference30 articles.
1. P. Aiena,
Fredholm and Local Spectral Theory II,
Lecture Notes in Math. 2235,
Springer, Cham, 2018.
2. R. Benjamin,
Spectral mapping theorems for the upper Weyl and upper Browder spectra,
Quaest. Math. 41 (2018), no. 7, 951–961.
3. R. Benjamin, N. J. Laustsen and S. Mouton,
r-Fredholm theory in Banach algebras,
Glasg. Math. J. 61 (2019), no. 3, 615–627.
4. M. Berkani,
Continuous Fredholm theory, regularities and semiregularities,
Complex Anal. Oper. Theory 15 (2021), no. 6, Paper No. 105.
5. M. Berkani and J. J. Koliha,
Weyl type theorems for bounded linear operators,
Acta Sci. Math. (Szeged) 69 (2003), no. 1–2, 359–376.