Weak Beurling Property and Extensions to Invertibility

Let E , E∗ be Hilbert spaces. Given a Hilbert space H of holomorphic functions in a domain Ω in C, consider the multiplier space MH(E , E∗). It is shown that for “nice enough” H, the following statements are equivalent for f ∈ MH(E , E∗): (1) There exists a g ∈ MH(E∗, E) such that g(z)f(z) = IE for all z ∈ Ω. (2) There exists a Hilbert space Ec and fc ∈ MH(Ec, E∗) such that F (z) := [ f(z) fc(z) ] : E ⊕ Ec → E∗ is invertible for every z ∈ Ω, that is, there exists a function G ∈ MH(E∗, E ⊕ Ec) such that G(z)F (z) = IE⊕Ec and F (z)G(z) = IE∗ for all z ∈ Ω. Moreover, we characterize the spaces H for which (1) and (2) above are equivalent. Since this characterization has a close relation with Beurling’s theorem for shift invariant subspaces of H, we call this property of H the weak Beurling property. We show that all reproducing kernel Hilbert spaces with complete Nevanlinna-Pick kernels have the weak Beurling property. This produces a large class of examples for which (1) and (2) are equivalent. We also give an example of another space with the weak Beurling property. CDAM Research Report LSE-CDAM-2009-05