Let A: X → X be a bounded operator on a separable complex Hilbert space X with an inner product $⟨·,·{⟩}_X$. For b, c ∈ X, a weak resolvent of A is the complex function of the form $⟨{(I-zA)}^{-1}b,c{⟩}_X$. We will discuss an equivalent condition, in terms of weak resolvents, for A to be similar to a restriction of the backward shift of multiplicity 1.

We prove that if some power of an operator is ergodic, then the operator itself is ergodic. The converse is not true.