On Bounded and Compact Composition Operators in Polydiscs

Abstract

Recently MacCluer and Shapiro [6] have characterized the compact composition operators in Hardy and weighted Bergman spaces of the disc, and MacCluer [5] has made an extensive study of these opertors in the unit ball of Cn. Angular derivatives and Carleson measures have played an essential role in these studies. In this article we study composition operators in poly discs and characterize those operators which are bounded or compact in Hardy and weighted Bergman spaces. In addition to Carleson measure theorems resembling those of [5], [6], we give necessary and sufficient conditions satisfied by the maps inducing bounded or compact composition operators. We conclude by considering the role of angular derivatives on the compactness question explicitly.

Composition operators on the polydisk induced by affine maps

“…This property is far from being true on the polydisk, even if Jafari has proven in [5] that continuity on A 2 β 1 (D n ) implies continuity on A 2 β 2 (D n ) for any β 2 β 1 . The converse does not hold.…”

Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs

“…As an immediate application of these characterizations we prove Proposition 1.3 and a weighted Fejer-Riesz inequality. Further application of these results to a large class of operators on these spaces are discussed elsewhere [5]. For reference we shall state the result of Chang here.…”

“…spaces, and classify those operators which are compact on the Hardy and weighted Bergman spaces in terms of Carleson-type conditions. We give two immediate applications of these results here, and a much broader class of applications elsewhere [5]. …”

Switched nonlinear systems in the Koopman operator framework: Toward a Lie-algebraic condition for uniform stability

“…The Koopman operator is defined here as the composition operator on H 2 (D n (ρ)) with symbol ϕ t (see e.g. [4,5,12]).…”