Invariant Subspaces of Operators on lp-Spaces

While the algebra of infinite matrices is more or less reasonable, the analysis is not. Questions about norms and spectra are likely to be recalcitrant. Each of the few answers that is known is considered a respectable mathematical accomplishment.P.R. Halmos [3, p. 24]A continuous operator T: X → X on a Banach space is quasinilpotent at a pointx0 whenever limn→∞||Tnx0||1/n = 0. Several results on the existence of invariant subspaces of operators which act on lp-spaces and are quasinilpotent at a non-zero point are obtained. For instance, it is shown that if an infinite positive matrix A = [aij] defines a continuous operator on an lp-space (1 ≤ p < ∞) and A is quasinilpotent at a positive vector, then for any bounded double sequence of complex numbers {wij: i,j = 1, 2, ... } the operator defined by the weighted infinite matrix [wijaij] has a non-trivial complemented invariant closed subspace.