Athanassios Beslikas: Composition Operators and Rational Inner Functions on the bidisc

Abstract:

Let \(\Phi \in \mathcal{O}(\mathbb{D}^2, \mathbb{D}^2)\) be a holomorphic self map of the bidisc \(\mathbb{D}^2\), and consider the weighted Bergman spaces
\[A_\beta^2(\mathbb{D}^2) = \left\{f \in \mathcal{O}(\mathbb{D}^2) : \int_{\mathbb{D}^2} |f(z_1, z_2))|^2 (1 - |z_1|^2)^\beta (1 - |z_2|^2)^\beta dV(z_1, z_2) < + \infty\right\}\]where \(\beta \ge -1\). In the papers of Bayart [1] and Kosiński [5] respectively, it was proved that if the symbol \(\Phi\) is \(\mathcal{C}^2\)−smooth on the closure of the bidisc, the composition operator \(C_\Phi : A_\beta^2(\mathbb{D}^2) \to A_\beta^2(\mathbb{D}^2)\) is bounded if and only if the derivative matrix \(d_\zeta \Phi\) is invertible for all \(\zeta \in \mathbb{T}^2\) such that \(\Phi(\zeta) \in \mathbb{T}^2\).

A natural question that emerges is how the situation differs if the symbol \(\Phi\) does not satisfy a smoothness condition, or even worse, if it has some kind of singularity on the bitorus \(\mathbb{T}^2\). Some interesting consequences appear when we consider a holomorphic self map \(\Phi = (\varphi, \psi)\) of the bidisc, when \(\varphi\), \(\psi\) are Rational Inner Functions. In the present talk, some recent results obtained in this direction will be presented.

Some references which are intimately connected to the main topic of this talk are [3] [4] and [6]. The talk is planned to be accessible for everyone who has a decent background into complex analysis (one dimensional setting is still fine) and basic notions of operator theory.

References
[1] F. Bayart, Composition operators on the polydisk induced by affine maps, Journal of Functional Analysis Volume 260, Issue 7, 1 April 2011, Pages 1969-2003.
[2] K. Bickel, Fundamental Agler Decompositions, Integral Equations and Operator Theory, Published: 11 September 2012 Volume 74, pages 233–257, (2012). , 1 December 2013, Pages 2753-2790
[3] K. Bickel, J.E. Pascoe, A. Sola, Derivatives of Rational Inner Functions and integrability at the boundary, Proceedings of the London Mathematical Society, Vol. 116, Issue 2, pp.281-329
[4] G. Knese, Rational Inner Functions in the Schur-Agler class of the polydisc, Publicacions Matemàtiques, Vol. 55, No. 2 (2011), pp. 343-357
[5] Ł. Kosiński, Composition operators on the polydisc, Journal of Functional Analysis, Volume 284, Issue 5, 1 March 2023, 109801.
[6] H. Koo, M. Stessin, K. Zhu, Composition operators on the polydisc induced by smooth symbols, Journal of Functional Analysis 254 (2008) 2911–2925