Norm closed invariant subspaces in H∞ and L∞

Abstract

We characterize norm closed subspaces B of L∞(∂D) such that C(∂D)B ⊂ B, and maximal ones in the family of proper closed subspaces B of L∞(∂D) such that A(D)B ⊂ B, where A(D) is the disk algebra. Analogously, we characterize closed subspaces of H∞ that are simultaneously invariant under S and S∗, the forward and the backward shift operators, and maximal invariant subspaces of H∞.