Backward shift operator, Dirichlet-type spaces, weighted Bergman spaces, cyclic function, noncyclic function, invariant subspaces, pseudocontinuation, Bergman shift operator
We study the backward shift operator on Hilbert spaces H_α (for α ≥ 0) which are norm equivalent to the Dirichlet-type spaces D_α. Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an elementary formula expressing the inner product on H_α in terms of a weighted superposition of backward shifts.
© 2005 American Mathematical Society
Garcia, S.R., The backward shift on Dirichlet-type spaces, Proc. Amer. Math. Soc. 133, no. 10, (2005), 3047–3056. MR2159784 (2006e:30038)