Weakly /^pstable linear operators are power stable

We prove that if a bounded linear operator A on a Banach space X is such that, for any x ε X and any y ε X*, the sequence ⟨A^k.x, y,⟩ is in; is in l^pwhere p ε (1, ∞), then the spectral radius of A is smaller than one. This solves the discrete-time version of a problem raised by Pritchard and Zabczyk (1983). As a consequence, if the linear time-invariant discrete-time systems associated with A are l^q-input-bounded state stable, where qε(1,∞), then A is power stable.