Maximal Invariant Subspaces For Hilbert Spaces Of Analytic Functions Over The Unit Disc

In this paper, we focus on maximal invariant subspaces for ordered analytic Hilbert spaces over the unit disc and the Dirichlet space. Firstly, we show that maximal in-variant subspaces in ordered analytic Hilbert spaces have index one. Secondly, we obtain that if M is an invariant subspace of a class of ordered analytic Hilbert spaces and dim M (?) zM=1,then every maximal invariant subspace of M is of the form N=(z-λ) M. Finally, we give a complete description for maximal invariant sub-spaces of the Dirichlet space. That is, an invariant subspace M of the Dirichlet space is maximal if and only if M is of the form M=[z-λ]=(z-λ)D, λ€D.