Invariant subspaces and rational approximation

Abstract

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 H 2( 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 H 2( 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.