Polynomially bounded and power-bounded operators have played an important role in the development of this area, and there are a number of interesting results, counterexamples, and open questions about these operators. In particular, we will present Foguel's example [98] of a power-bounded operator and Pisier's example [191] of a polynomially bounded operator that are not similar to contractions.

Recall that an operator T is power-bounded provided that there is a constant M such that ||T n|| ≤ M for all n ≥ 0. Clearly, if T = S-1CS with C a contraction, then T is power-bounded with ||T n|| ≤ ||S-1|| ||S||.

It is fairly easy to see (Exercise 10.1), by using the Jordan form, that a matrix T ∈ Mn is power-bounded if and only if it is similar to a contraction. Sz.-Nagy [229] proved that the same characterization holds when T is a compact operator. This led naturally to the conjecture that an arbitrary operator is similar to a contraction if and only if it is power-bounded. Foguel provided the first example of a power-bounded operator that is not similar to a contraction.

Recall that an operator is polynomially bounded provided there is a constant K such that ||p(T)|| ≤ K||p||∞ for every polynomial p, where the ∞-norm is the supremum norm over the unit disk.