Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 38, 2013, 547-564

ESTIMATES FOR APPROXIMATION NUMBERS OF SOME CLASSES OF COMPOSITION OPERATORS ON THE HARDY SPACE

Daniel Li, Hervé Queffélec and Luis Rodríguez-Piazza

Université Lille Nord de France, Université d'Artois, Faculté des Sciences Jean Perrin
Laboratoire de Mathématiques de Lens EA 2462 & Fédération CNRS Nord-Pas-de-Calais FR 2956
Rue Jean Souvraz, S.P. 18, F-62 300 Lens, France; daniel.li 'at' euler.univ-artois.fr

Université Lille Nord de France, Université Lille 1, Sciences et Technologies
Laboratoire Paul Painlevé U.M.R. CNRS 8524 & Fédération CNRS Nord-Pas-de-Calais FR 2956
F-59 655 Villeneuve d'Ascq Cedex, France; Herve.Queffelec 'at' univ-lille1.fr

Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS
Apartado de Correos 1160, 41 080 Sevilla, Spain, piazza 'at' us.es

Abstract. We give estimates for the approximation numbers of composition operators on H², 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^{-cn/log n}, very near to the minimal value e^-cn. We also see the limitations of our methods. To finish, we improve a result of El-Fallah, Kellay, Shabankhah and 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 : H² \to H², which is in all Schatten classes, and such that \phi = 1 on K and |\phi| < 1 outside K.

2010 Mathematics Subject Classification: Primary 47B06; Secondary 30J10, 47B33.

Key words: Approximation numbers, Blaschke product, composition operator, cusp map, Hardy space, modulus of continuity, Schatten classes.

Reference to this article: D. Li, H. Queffélec and L. Rodríguez-Piazza: Estimates for approximation numbers of some classes of composition operators on the Hardy space. Ann. Acad. Sci. Fenn. Math. 38 (2013), 547-564.

Full document as PDF file

doi:10.5186/aasfm.2013.3823