Distributionally n-Scrambled Set for Weighted Shift Operators

Abstract

The aim of the present paper is to investigate distributionally n-scrambled sets for weighted shift operators. We prove that the unilateral weighted shift operator admits densely invariant distributionally n-e-scrambled linear manifolds for any e ∈ (0, 1) and any integer n ⩾ 2, showing that this operator can exhibit maximal distributional n-chaos on a dense invariant linear manifold. Analogous results for the bilateral weighted shift operator are also obtained.