Eigenvalues and hypercyclicity in omega
It is well known that an infinite matrix \(A=(a_{i,j})\) defines a continuous linear operator on the Fréchet space \(\omega\) of all sequences if and only if each row \(i\) has a finite number of coordinates different from zero. Letting \(n_i\) denote the maximum \(j\) such that \(a_{i,j} \neq 0\), the author defines upper staircase matrices as those matrices \(A\) defining a continuous operator on \(\omega\) such that the sequence \((n_i)\) is strictly increasing and \(i < n_i\) for each row \(i\). The set \(S\) of upper staircase matrices is a noncommutative semigroup. It is proved that upper staircase matrices define continuous linear operators on \(\omega\) that have many eigenvalues and eigenvectors, that are hypercyclic, and even chaotic in the complex case. Extending a theorem by Bès and Conejero, the author proves that countably many strictly upper triangular matrices on \(\omega\) which are also upper staircase have a common hypercyclic subspace. The article concludes with a list of open questions.
scientific article; zbMATH DE number 6112715
scientific article