The Coburn–Simonenko Theorem for Toeplitz Operators Acting Between Hardy Type Subspaces of Different Banach Function Spaces

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Abstract

Let Γ be a rectifiable Jordan curve, let X and Y be two reflexive Banach function spaces over Γ such that the Cauchy singular integral operator S is bounded on each of them, and let M(X, Y) denote the space of pointwise multipliers from X to Y. Consider the Riesz projection P= (I+ S) / 2 , the corresponding Hardy type subspaces PX and PY, and the Toeplitz operator T(a) : PX→ PY defined by T(a) f= P(af) for a symbol a∈ M(X, Y). We show that if X↪ Y and a∈ M(X, Y) \ { 0 } , then T(a) ∈ L(PX, PY) has a trivial kernel in PX or a dense image in PY. In particular, if 1 < q≤ p< ∞, 1 / r= 1 / q- 1 / p, and a∈ Lr≡ M(Lp, Lq) is a nonzero function, then the Toeplitz operator T(a), acting from the Hardy space Hp to the Hardy space Hq, has a trivial kernel in Hp or a dense image in Hq.

Original languageEnglish
Article number91
JournalMediterranean Journal of Mathematics
Volume15
Issue number3
DOIs
Publication statusPublished - 1 Jun 2018

Keywords

  • Banach function space
  • Coburn–Simonenko theorem
  • Pointwise mutiplier
  • Symbol
  • Toeplitz operator
  • Variable Lebesgue space

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