Banach algebras of structured matrix sequences
In the first part of this paper, the authors present a general theory about the asymptotic behavior of operator sequences. Given a Banach space \(X\) and an approximate projection \({\mathcal P}\) (a bounded sequence of operators in the algebra of all bounded and linear operators on \(X\) with certain properties), the authors introduce the notions of operator \({\mathcal P}\)-compact, operator \({\mathcal P}\)-Fredholm and \({\mathcal P}\)-strong convergence of a sequence of operators. Then they establish a Fredholm property for structured sequences in Banach algebras of the type \({\mathcal F}^T\) (certain \(C^*\)-algebras of bounded sequences of operators acting on finite-dimensional spaces) and prove a stability result. In the second part, the authors apply the general theory to the space \(\ell^p (\mathbb{Z}^K,\mathbb{C}^N)\), where \(K,N \in \mathbb{N}\).
scientific article; zbMATH DE number 5505626
scientific article