https://hal-univ-artois.archives-ouvertes.fr/hal-00704746Li, DanielDanielLiLML - Laboratoire de Mathématiques de Lens - UA - Université d'ArtoisQueffélec, HervéHervéQueffélecLPP - Laboratoire Paul Painlevé - Université de Lille - CNRS - Centre National de la Recherche ScientifiqueRodriguez-Piazza, LuisLuisRodriguez-PiazzaDepartamento de Analisis MatematicoEstimates for approximation numbers of some classes of composition operators on the Hardy spaceHAL CCSD2012approximation numbersBlaschke productcomposition operatorcusp mapHardy spacemodulus of continuitySchatten classes[MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA]Li, Daniel2012-06-06 10:54:572023-03-24 14:52:552012-06-06 12:59:02enPreprints, Working Papers, ...https://hal-univ-artois.archives-ouvertes.fr/hal-00704746/documentapplication/pdf1We 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 $K$.