Composition Operators on a Class of Analytic Function Spaces Related to Brennan’s Conjecture

Abstract

Abstract. Brennan's conjecture in univalent function theory states that if τ is any analytic univalent transform of the open unit disk D onto a simply connected domain G and −1/3 < p < 1, then 1/(τ ) p belongs to the Hilbert Bergman space of all analytic square integrable functions with respect to the area measure. We introduce a class of analytic function spaces L 2 a (µp) on G and prove that Brennan's conjecture is equivalent to the existence of compact composition operators on these spaces for every simply connected domain G and all p ∈ (−1/3, 1). Motivated by this result, we study the boundedness and compactness of composition operators in this setting.

A conformal mapping example

“…There some classes of G for which it is solved, for example, in [1] Brennan's conjecture is proven for any G which is a component of Fatou's set of any second degree polynomial. In [6], V. Matache and the first author gave an equivalent formulation of Brennan's Conjecture in terms of composition operators acting on certain Hilbert spaces of analytic functions on G.…”

“…There are interesting and promising reformulations of Brennan's conjecture, see, for example, [4] or [6]. In particular, it was shown in [6] that Brennan's Conjecture is equivalent to the statement that for each p ∈ (−1/3, 1) and simply connected domain G, there is an analytic selfmap τ of G such that C τ is a compact operator on L 2 a (μ p ).…”

“…Un progrès important a été réalisé dans [4]. On trouvera dans [6] une formulation équivalente de la Conjecture de Brennan en termes d'opérateurs de composition agissant sur certains espaces de Hilbert de fonctions analytiques sur G.…”