Multipliers between model spaces

Emmanuel Fricain (1) , Andreas Hartmann (2) , William T. Ross (3)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 IMB - Institut de Mathématiques de Bordeaux
3 Department of Mathematics

In this paper we characterize the multipliers from one model space (of the disk) to another. Our characterization involves kernels of Toeplitz operators and Carleson measures. We illustrate this characterization in different situations and in large classes of examples. As it turns out, under certain circumstances, every multiplier between the two model spaces is a bounded function. However, this is not always the case. In the case of onto multipliers, this answers a question posed by Crofoot. When considering model spaces of the upper-half plane, we will discuss in some detail when the associated inner function is a meromorphic inner function. This connects to de Branges spaces of entire functions which are closely related to different important problems in complex analysis (e.g., zero distribution, differential equations, and completeness problems). When the derivative of the associated inner function is bounded, we show that the set of multipliers contains the kernel of an associated Toeplitz operator.

Mots clés: Hardy spaces; inner functions; model spaces; multipliers; de Branges Rovnyak spaces

Domaines: Variables complexes [math.CV]

Emmanuel Fricain, Andreas Hartmann, William T. Ross. Multipliers between model spaces. 2016. ⟨hal-01319621v2⟩