On Centered and Weakly Centered Operators

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 ≥ 1 which satisfies (1) [resp., (1) restricted]. This number will be called the polynomial bound [resp., the power bound] of T and will be denoted by Mpb(T ) [resp., Mpw(T )]. (If T / ∈ (PB) [resp., T / ∈ (PW)], we set Mpb [resp., Mpw] equal to +∞.) 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 ≥ 1 such that one has