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Cyclic vectors and invariant subspaces for the backward shift operator

R. G. DouglasH. S. ShapiroA. L. Shields — 1970

Annales de l'institut Fourier

The operator U of multiplication by z on the Hardy space H 2 of square summable power series has been studied by many authors. In this paper we make a similar study of the adjoint operator U * (the “backward shift”). Let K f denote the cyclic subspace generated by f ( f H 2 ) , that is, the smallest closed subspace of H 2 that contains { U * n f } ( n 0 ) . If K f = H 2 , then f is called a cyclic vector for U * . Theorem : f is a cyclic vector if and only if there is a function g , meromorphic and of bounded Nevanlinna characteristic...

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