Resolvent criteria for similarity to a normal operator with spectrum on a curve

We give some new criteria for a Hilbert space operator with spectrum on a smooth curve to be similar to a normal operator, in terms of pointwise and integral estimates of the resolvent. These results generalize criteria of Stampfli, Van Casteren and Naboko, and answers several questions posed by Stampfli in [48]. The main tools are from our recent results [12] on dilation to the boundary of the spectrum, along with the Dynkin functional calculus for smooth functions, which is based on pseudoanalytic continuation.