## 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.

Original language | English |
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Pages (from-to) | 285-309 |

Number of pages | 25 |

Journal | Journal of Functional Analysis |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1971 |

### Bibliographical note

Funding Information:supported by N.S.F. grant GP-7653.

Copyright:

Copyright 2014 Elsevier B.V., All rights reserved.

## ASJC Scopus subject areas

- Analysis