STRONGLY SPLITTING WEIGHTED SHIFT OPERATORS ON BANACH SPACES AND UNICELLULARITY

We introduce the notion of strong splitting operator on a separable Banach space, and prove a structure theorem for this operator. We consider the weighted shift operator T, Te-n = lambda(n)e(n+1), n >= 0, acting in the Banach space X with basis {e(n)}(n >= 0). We give some sufficient conditions for X and for the weight sequence {lambda(n)}(n >= 0) under which the operator is unicellular, that is, every nontrivial invariant subspace E of T has the form E = X-i := Span {e(k) : k >= i} for some i >= 1; and prove that the restricted operators T vertical bar X-i (i >= 1) are strong splitting. Moreover, we describe in terms of so-called discrete Duhamel operator and diagonal operator all extended eigenvectors of the operators T vertical bar X-i (i >= 1).