NEW COUNTEREXAMPLES ON RITT OPERATORS, SECTORIAL OPERATORS AND R-BOUNDEDNESS

Let D be a Schauder decomposition on some Banach space X. We prove that if D is not R-Schauder, then there exists a Ritt operator T is an element of B( X) which is a multiplier with respect to D such that the set {T-n : n >= 0} is not R-bounded. Likewise, we prove that there exists a bounded sectorial operator A of type 0 on X which is a multiplier with respect to D such that the set {e(-tA) : >= 0} is not R-bounded.