Invariant subspaces and rational approximation

Let μ be a positive measure of compact support in the complex plane. Let P be the set of complex polynomials and R_μ the set of rational functions having no poles on the support of μ. For each p, 1 ≤ p < ∞, let L^p(dμ) have its usual meaning. Denote by H^p(dμ) and R^p(dμ) the closures in L^p(dμ) of P and R_μ respectively. The principal aim of this paper is to establish, in certain cases, the existence of nontrivial closed subspaces in H^p(dμ) and R^p(dμ) which remain invariant under multiplication by P and R_μ. If p > 2 it is shown that R^p(dμ) always has an R_μ-invariant subspace. Specifically, if p > 2 either R^p(dμ) = L^p(dμ) or R^p(dμ) has an R_μ-invariant subspace of finite codimension. An example is provided to indicate that this dichotomy need not persist when p = 2. H^p(dμ) is similar to R^p(dμ) in that it always has a P-invariant subspace when p > 2. Concerning H^p(dμ) particular attention is given to the case where μ is Lebesgue measure dx dy restricted to a compact set E. In this connection it is shown that H^p(E, dx dy) has a P-invariant subspace whenever p ≠ 2 and that H²(E, dx dy) has also, provided E has "finite perimeter." In a note added in the proof the author claims to have established for an arbitrary compact E the existence of P-invariant subspaces in H²(E, dx dy). A number of results concerning approximation by polynomials and rational functions in the L^p(E, dx dy) norm are obtained as by-products of this investigation. Some of these were obtained earlier by S. O. Sinanjan while others extend his work.