2012/06/06 by Daniel Li, Li, Daniel, Hervé Queffélec +3 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · doi:10.48550/arxiv.1206.1179
openalex publication_date 2012/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 √ 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 Cϕ\colon H2 → H2, which is in all Schatten classes, and such that ϕ= 1 on K and |ϕ| < 1 outside K.