Invariant subspaces and rational approximation

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Abstract

Let μ be a positive measure of compact support in the complex plane. Let P be the set of complex polynomials and Rμ the set of rational functions having no poles on the support of μ. For each p, 1 ≤ p < ∞, let Lp(dμ) have its usual meaning. Denote by Hp(dμ) and Rp(dμ) the closures in Lp(dμ) of P and Rμ respectively. The principal aim of this paper is to establish, in certain cases, the existence of nontrivial closed subspaces in Hp(dμ) and Rp(dμ) which remain invariant under multiplication by P and Rμ. If p > 2 it is shown that Rp(dμ) always has an Rμ-invariant subspace. Specifically, if p > 2 either Rp(dμ) = Lp(dμ) or Rp(dμ) has an Rμ-invariant subspace of finite codimension. An example is provided to indicate that this dichotomy need not persist when p = 2. Hp(dμ) is similar to Rp(dμ) in that it always has a P-invariant subspace when p > 2. Concerning Hp(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 Hp(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 Lp(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 languageEnglish
Pages (from-to)285-309
Number of pages25
JournalJournal of Functional Analysis
Volume7
Issue number2
DOIs
StatePublished - 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

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