Service interruption on Monday 11 July from 12:30 to 13:00: all the sites of the CCSD (HAL, Epiciences, SciencesConf, AureHAL) will be inaccessible (network hardware connection).
Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

Estimates for approximation numbers of some classes of composition operators on the Hardy space

Abstract : We give estimates for the approximation numbers of composition operators on $H^2$, in terms of some modulus of continuity. For symbols whose image is contained in a polygon, we get that these approximation numbers are dominated by $\e^{- c \sqrt n}$. When the symbol is continuous on the closed unit disk and has a domain touching the boundary non-tangentially at a finite number of points, with a good behavior at the boundary around those points, we can improve this upper estimate. A lower estimate is given when this symbol has a good radial behavior at some point. As an application we get that, for the cusp map, the approximation numbers are equivalent, up to constants, to $\e^{- c \, n / \log n }$, very near to the minimal value $\e^{- c \, n}$. We also see the limitations of our methods. To finish, we improve a result of O. El-Fallah, K. Kellay, M. Shabankhah and H. Youssfi, in showing that for every compact set $K$ of the unit circle $\T$ with Lebesgue measure $0$, there exists a compact composition operator $C_\phi \colon H^2 \to H^2$, which is in all Schatten classes, and such that $\phi = 1$ on $K$ and $|\phi | < 1$ outside $K$.
Document type :
Preprints, Working Papers, ...
Complete list of metadata

Cited literature [22 references]  Display  Hide  Download
Contributor : Daniel Li Connect in order to contact the contributor
Submitted on : Wednesday, June 6, 2012 - 10:54:57 AM
Last modification on : Wednesday, March 23, 2022 - 3:50:06 PM
Long-term archiving on: : Friday, September 7, 2012 - 2:27:01 AM


Files produced by the author(s)


  • HAL Id : hal-00704746, version 1
  • ARXIV : 1206.1179



Daniel Li, Hervé Queffélec, Luis Rodriguez-Piazza. Estimates for approximation numbers of some classes of composition operators on the Hardy space. 2012. ⟨hal-00704746⟩



Record views


Files downloads