Circles in the spectrum and numerical ranges

By a classical result of Arveson, the spectrum of a unitary operator U is the entire unit circle if and only if for every n there exists a non-zero vector x such that x, Ux,..., U^nx are mutually orthogonal. We will show that this result can be essentially generalized to the situation when the essential approximate point spectrum of a (not necessarily unitary) operator contains the unit circle. The approach can be applied in other situations. For example it implies that the weak convergence of operator powers implies the uniform

convergence of their compressions to an infinite-dimensional subspace.