A Beurling type theorem in weighted Bergman spaces

Abstract

For the vector-valued Hardy space H2(U) and the standard weighted Bergman space An(Y) with coefficient Hilbert spaces U and Y, we single out a class of contractive multipliers from H2(U) to An(Y) which we call partially isometric multipliers. We then show that a closed subspace M⊂An(Y) is invariant under the shift operator Sn:f(z)↦zf(z) if and only if M=Φ⋅H2(U) for some partially isometric multiplier Φ from H2(U) to An(Y).