Stellenbosch University
Perturbation ideals and Fredholm theory in Banach
algebras
Tshikhudo Lukoto\(^*\), University of Limpopo
Heinrich Raubenheimer, University of
Johannesburg
SAMS Subject Classification Number: 11
In 1971, Lebow and Schechter in [1] introduced the notion of perturbation classes where one of the important result discovered is that the perturbation of \(\mathcal{A}^{-1}\), the group of invertible elements in a Banach algebra \(\mathcal{A}\), is equal to Rad\((\mathcal{A})\), the radical in \(\mathcal{A}\), and we denoted this result as \[\mathcal{P}(\mathcal{A}^{-1})=\mbox{Rad}(\mathcal{A}).\qquad{(1)}\] We use this notion to characterize perturbation ideals of sets that generate the familiar spectra in Fredholm theory. At first we classify the set in question as either regularity or semiregularity, a concept which is set out in [2].
References
[1] A. Lebow and M. Schechter, Semigroups of Operators and Measures of Noncompactness, Journal of Functional Analysis 7 (1971), 1-26.
[2] V. M\(\ddot{u}\)ller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Operator Theory Advances and Applications, Vol. 139, Birkh\(\ddot{a}\)user Verlag, Basel-Boston-Berlin, 2007.
