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Approximation numbers of composition operators on the Dirichlet space. (English) Zbl 1309.47022

Summary: We study the decay of approximation numbers of compact composition operators on the Dirichlet space. We give upper and lower bounds for these numbers. In particular, we improve on a result of O. El-Fallah et al. [J. Funct. Anal. 260, No. 6, 1721–1733 (2011; Zbl 1214.47025)], on the set of contact points with the unit circle of a compact symbolic composition operator acting on the Dirichlet space \(\mathcal{D}\). We extend their results in two directions: first, the contact only takes place at the point \(1\). Moreover, the approximation numbers of the operator can be arbitrarily subexponentially small.

MSC:

47B33 Linear composition operators
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
30H20 Bergman spaces and Fock spaces

Citations:

Zbl 1214.47025
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References:

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