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

Abstract

We give examples of composition operators C Φ on H 2 (D 2 ) showing that the condition Φ ∞ = 1 is not sufficient for their approximation numbers a n (C Φ ) to satisfy lim n→∞ [a n (C Φ )] 1/ √ n = 1, contrary to the 1-dimensional case.We also give a situation where this implication holds. We make a link with the Monge-Ampère capacity of the image of Φ.

Approximation numbers of weighted composition operators

“…Another application of weighted composition operators to the study of composition operators on spaces of several complex variables can be found in[29].…”

Pluricapacity and approximation numbers of composition operators

“…In any case, this lets us suspect that the formula of Theorem 4.1 holds in much more general cases. This is not quite true, as evidenced by our counterexample of [41,Theorem 5.12]. Nevertheless, in good cases, this formula holds, as we will see in the next sections.…”

“…In [41], we pursued that line of investigation in dimension N ≥ 2, namely on H 2 (D N ), and showed that in some cases the implication (1.2) still holds ([41, Theorem 3.1]):…”

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

“…It is coined in [1] (see also [13] and [14]) that β ± N (C Φ ) are the suitable parameters for the composition operators on H 2 (D N ), and it is proved, for any N ≥ 1, that β − N (C Φ ) > 0, as soon as Φ is non degenerate (i.e. the Jacobian J Φ is not identically 0) and the operator C Φ is bounded on H 2 (D N ).…”

“…the Jacobian J Φ is not identically 0) and the operator C Φ is bounded on H 2 (D N ). As for an expression of β ± N (C Φ ) in terms of "capacity", only partial results are known so far ( [13] and [14]) and the application to a result like (1.1) fails in general. We gave an example of such a phenomenon in [13,Theorem 5.12].…”