On Invariant Subspaces for the Shift Operator

In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace M in the Hardy space Hp(D) (1 <= p<infinity) is invariant under the shift operator Mz on Hp(D) if and only if it is hyperinvariant under Mz, and that a closed linear subspace M in the Lebesgue space L2(partial derivative D) is reducing under the shift operator Mei theta on L2(partial derivative D) if and only if it is hyperinvariant under Mei theta. At the same time, we show that there are two large classes of invariant subspaces for Mei theta that are not hyperinvariant subspaces for Mei theta and are also not reducing subspaces for Mei theta. Moreover, we still show that there is a large class of hyperinvariant subspaces for Mz that are not reducing subspaces for Mz. Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space H2(D), which are the analogue of the formula of the reproducing function in the Bergman space A2(D). In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces.