Noncompactness of Toeplitz operators between abstract Hardy spaces

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Abstract

In the beginning of 1960s, Brown and Halmos proved that a Toeplitz operator T(a) is compact on the Hardy space H2= H[L2] over the unit circle T if and only if a= 0 a.e. Recently, Leśnik [13] generalized this result to the setting of Toeplitz operators acting between abstract Hardy spaces H[X] and H[Y] built upon possibly different rearrangement-invariant Banach function spaces X and Y over T such that Y has nontrivial Boyd indices. We show that the general principle of noncompactness of nontrivial Toeplitz operators between abstract Hardy spaces H[X] and H[Y] remains true for much more general spaces X and Y. In particular, there are no nontrivial compact Toeplitz operators on the Hardy space H1= H[L1] , although L1 has trivial Boyd indices.

Original languageEnglish
Article number29
JournalAdvances in Operator Theory
Volume6
Issue number2
DOIs
Publication statusPublished - Apr 2021

Keywords

  • Abstract Hardy space
  • Compactness
  • Pointwise multiplier
  • Toeplitz operator

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