Algebras of Convolution Type Operators with Continuous Data do Not Always Contain All Rank One Operators

Alexei Karlovich, Eugene Shargorodsky

Research output: Contribution to journalArticlepeer-review

Abstract

Let X(R) be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X(R). The algebra CX(R˙) of continuous Fourier multipliers on X(R) is defined as the closure of the set of continuous functions of bounded variation on R˙ = R∪ { ∞} with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author [11] that if the space X(R) is reflexive, then the ideal of compact operators is contained in the Banach algebra AX(R) generated by all multiplication operators aI by continuous functions a∈ C(R˙) and by all Fourier convolution operators W(b) with symbols b∈ CX(R˙). We show that there are separable and non-reflexive Banach function spaces X(R) such that the algebra AX(R) does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces Lp,1(R) with 1 < p< ∞.

Original languageEnglish
Article number16
JournalIntegral Equations And Operator Theory
Volume93
Issue number2
DOIs
Publication statusPublished - Apr 2021

Keywords

  • Algebra of convolution type operators
  • Continuous Fourier multiplier
  • Hardy-Littlewood maximal operator
  • Lorentz space
  • Rank one operator
  • Separable Banach function space

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