Beurling-type Theorems On Vector-valued Hardy Space And Vector-valued Dirichlet Space

Let E be a separable Hilbert space, II2{E) and (?)(E) denote vector-valued Hardy space and vector-valued Dirichlet space whoes values are in E, M is a nonzero subspace of Ⅱ2(E) or (?)(E). If α∈E, f∈M, a*f is the Hadamard product of a and f. Define E*M={a*f:a∈E, f∈M}. In this paper, we completely characterized the nonzero shift invariant subspace M which satisfy E*m(?)M in vector-valued Hardy space and vector-valued Dirichlet space.