Power-bounded invertible operators and invertible isometries on Lp spaces

Abstract

It is shown that an invertible isometry on lp, where 1 ≤ p < ∞ and p ¹ 2, is a scalar-type spectral operator provided its spectrum is a proper subset of the unit circle. A similar, though weaker, analysis is also considered for invertible isometries on more general Lp spaces. These results are used to give several examples of invertible operators U on Lp spaces, where p ∈ (1,∞) and p ¹ 2, such that supn∈Z ½½Un½½ < ∞ but U is not similar to an invertible isometry. This contrasts with the situation on Hilbert space, where the condition sup n∈Z ½½Un½½ < ∞ on an invertible operator U implies that U is similar to a unitary operator.