Invariant subspaces of operators on \(\ell_ p\)-spaces
The authors prove interesting, and potentially useful, invariant subspace results for positive operators on \(\ell^ p\) that are quasinilpotent at a non-zero positive point \(x_ 0\) (i.e., \(\lim_{n\to\infty} \| T^ n x_ 0\|^{1/n}= 0\) and all entries \(t_{ij}\) of the matrix for \(T\) and all coordinates of \(x_ 0\) are non-negative). Such an operator must have a non-trivial complemented closed invariant subspace. More generally, they prove that for such a \(T\) and for any bounded double sequence of complex numbers \(\{w_{ij}\}\), the operator defined by the weighted infinite matrix \([w_{ij} t_{ij}]\) has a non-trivial invariant subspace.
quasinilpotent at a non-zero positive point
non-trivial complemented closed invariant subspace
scientific article; zbMATH DE number 515236