On Polynomially Bounded Weighted Shifts

(1) ‖p(T )‖ ≤M sup{|p(ζ)| : |ζ| = 1} ∀ polynomial p, and to be power bounded (notation T ∈ (PW)) if (1) holds for every polynomial of the special form p(ζ) = ζ where n is a positive integer. If T ∈ (PB) [resp., T ∈ (PW)], then there is a smallest number M which satisfies (1) [resp., (1) restricted]. This number will be called the polynomial bound of T [resp., the power bound of T ] and denoted by Mpb(T ) [resp., Mpw(T )]. Also an operator T in L(H) is said to be similar to a contraction (notation: T ∈ (SC)) if there exists an invertible operator S in L(H) such that ‖S−1TS‖ ≤ 1, and to be completely polynomially bounded (notation: T ∈ (CPB)) if there exists an M > 0 such that one has