Polynomially Riesz elements: a study via Fredholm theory
An important concept in the spectral theory of Banach algebras is that of a quasinilpotent element: an element \(a \in A\) for which the spectrum is \({0}\), that is, \(\{\lambda \in \mathbb{C}\mid \lambda 1_A − a \notin A^{-1} \} = \{0\}\). Let \(T : A \to B\) be a Banach algebra homomorphism. Then an element \(a \in A\) is called Riesz (relative to \(T\) ) if \(T a\) is quasinilpotent. The aim of this talk is to discuss the following generalisation of Riesz elements, which was introduced in [1], and developed further in [2]: an element \(a \in A\) is called polynomially Riesz (relative to \(T\)) if there exists a non-zero polynomial \(p\) such that \(p(a)\) is Riesz, i.e. \(\sigma (Tp(a)) = \{0\}\).
We will investigate these concepts through the Fredholm theory of Banach algebras, which studies different concepts (relative to Banach algebra homomorphisms) that are related to invertibility. We show that quasinilpotent and Riesz elements form an important part of the Fredholm theory of Banach algebras. Then we investigate which of their properties have analogues where we can replace Riesz elements with polynomially Riesz elements. We will focus on the perturbation and spectral properties involving Fredholm, Weyl and Browder elements and spectra.
References:
- SČ Živković-Zlatanović, DS Djordjević, R Harte, Ruston, Riesz and perturbation classes, J. Math. Anal. Appl., Vol. 389, 2012, 871–886.
- SČ Živković-Zlatanović, R Harte, Polynomially Riesz elements, Quaest. Math., Vol. 38, 2015, 573–586.