Contractions and hyperdistributions with carleson set spectrum

Let omega = (omega(n))(n greater than or equal to 1) be a log concave seque nce such that lim inf(n-->+infinity) omega(n)/n(e) > 0 for some c > 0 and ( (log omega(n))/n(alpha))(n greater than or equal to 1) is nonincreasing for some alpha < 1/2. We show that, if T is a contraction on the Hilbert space with spectrum a Carleson set, and if \\T-n\\ = O(omega(n)) as n tends to infinity with Sigma(n greater than or equal to 1) l/(nlog omega(n)) = + in finity, then T is unitary. On the other hand, if Sigma(n greater than or eq ual to 1) l/(n log omega(n)) < + infinity, then there exists a (non-unitary ) contraction T on the Hilbert space such that the spectrum of T is a Carle son set, \\T-n\\ = O(omega(n)) as n tends to +infinity, and lim sup(n-->+in finity) \\T-n = + infinity.