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

Abstract

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 √ 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 φ : H 2 → H 2 , which is in all Schatten classes, and such that φ = 1 on K and |φ| < 1 outside K.

Approximation numbers of weighted composition operators

“…The definition of a n (T ) also makes sense for T : X → Y an operator between Banach spaces (see Theorem 2.8 to come). Coming back to the hilbertian setting, two other useful alternative definitions (respectively in terms of Bernstein and Gelfand numbers) are, denoting by S E the unit sphere of a subspace E of H (see [11,Chapter 2], or [27]):…”

“…More recently, also the membership of C ϕ in various smaller ideals I of bounded operators on H (such as the p-Schatten class), and more precisely the behavior of the approximation numbers a n (C ϕ ) of C ϕ , was studied in depth in several papers (see e.g. [27,25,26,28]).…”

An extremal composition operator on the Hardy space of the bidisk with small approximation numbers

“…It was often used by the authors ( [11], [8]) as an extremal example. We first recall the definition of χ.…”

Some examples of composition operators and their approximation numbers on the Hardy space of the bidisk

“…As above, we have sup k>K T k ≤ c K . For the cusp map, we have a n (C χ ) e −αn/ log n ( [20], Theorem 4.3); hence a n (T k ) e −αn/ log n . We take n 0 = n 1 = · · · = n K = K [log K] (where [log K] is the integer part of log K).…”