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

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Daniel Li
Hervé Queffélec
  • Function : Author
  • PersonId : 916881
Luis Rodriguez-Piazza
  • Function : Author
  • PersonId : 859619

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 $K$.
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Dates and versions

hal-00704746 , version 1 (06-06-2012)

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