In this talk we will consider the approximation numbers of differences of composition operators acting on the Hardy-Hilbert space $H^2(\mathbb{D})$. The component structure of bounded composition operators is a widely studied area and in order to understand whether two composition operators belong to the same component, it is important to understand how their difference behaves (compact, bounded etc.). One of the key elements in understanding the behavior of an operator is to consider its approximation numbers since it gives us the information about how much our operator differs from a bounded/compact one. During the talk we will mention how we can combine these two topics in operator theory and how one can obtain optimal upper and lower bounds for approximation numbers of differences using classical invariants like Bernstein and Gelfand numbers and specific choices of Blaschke products from the underlying function space. 

Joint work with Frédéric Bayart of Laboratoire de Mathématiques Blaise Pascal.

References

[1] G. Lechner, D. Li, H. Queff ́elec, L. Rodriguez-Piazza : Approximation numbers of weighted composition operators. Journal of Functional Analysis 274, 1928–1958 (2018).

[2] J. Moorhouse, C. Toews : Differences of composition operators. Contemporary Mathematics 321, 207–213 (2003).

[3] H. Queff ́elec, K. Seip : Decay rates for approximation numbers of composition operators. Journal d’Analyse Math ́ematique 125, 371–399 (2015).