We characterize the contractions that are similar to the backward shift in the Hardy space $H^2$. This characterization is given in terms of the geometry of the eigenvector bundles of the operators.
Hyun-Kyoung Kwon. Sergei Treil. "Similarity of Operators and Geometry of Eigenvector Bundles." Publ. Mat. 53 (2) 417 - 438, 2009.