Power-bounded Invertible Operators and Invertible Isometries on Lp Spaces

It is shown that an invertible isometry on l(p), where 1 <= p < infinity and p not equal 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 L-p spaces. These results are used to give several examples of invertible operators U on L-p spaces, where p. (1, infinity) and p not equal 2, such that sup(n is an element of Z) parallel to U-n parallel to < infinity but U is not similar to an invertible isometry. This contrasts with the situation on Hilbert space, where the condition sup(n is an element of Z) parallel to U-n parallel to < infinity on an invertible operator U implies that U is similar to a unitary operator.