Composition operators with surjective symbol and small approximation numbers

Abstract

We give a new proof of the existence of a surjective symbol whose associated composition operator on \(H^2 (\mathbb{D})\) is in all Schatten classes, with the improvement that its approximation numbers can be, in some sense, arbitrarily small. We show, as an application, that, contrary to the \(1\)-dimensional case, for \(N \geq 2\), the behavior of the approximation numbers \(a_n = a_n (C_\phi)\), or rather of \(\beta^-_N = \liminf_{n \to \infty} [a_n]^{1/ n^{1/ N}}\) or \(\beta^+_N = \limsup_{n \to \infty} [a_n]^{1/ n^{1/ N}}\), of composition operators on \(H^2 (\mathbb{D}^N)\) cannot be determined by the image of the symbol.