# Adem Limani: Approximation problems in model spaces and beyond

**Time: **
Wed 2022-05-11 13.15 - 14.15

**Location: **
Kräftriket, House 6, Room 306 and Zoom

**Video link: **
Meeting ID: 692 1892 7142

**Lecturer: **
Adem Limani (Lunds universitet)

### Abstract

The model spaces are the invariant subspaces for the backward shift operator on the Hardy space \(H^2\), where the label "model space" stems from the classical theory of Sz.-Nagy and Foias, which says that any contractive and completely non-unitary linear operator on a Hilbert space can be modeled by the backward shift on a certain model space. Besides their intrinsic operator theoretical nature, these spaces also enjoy some very subtle function theoretical properties. For instance, a classical theorem on approximations in model spaces by A. B. Alek- sandrov says that functions in a model space which extend contin- uously to the boundary form a dense subset, despite the fact that in many instances, it is very difficult to construct even a single such func- tion. In this talk, we shall investigate the mechanisms which determine when classes of functions enjoying certain regularity properties on the boundary, form a dense subset in the model spaces. It turns out these problems can be reformulated in terms of Beurling-type theorems. We shall also explore some extensions of these problems to the setting of de Branges-Rovnyak spaces and illustrate some recently established connections to the theory of cyclic subnormal operators. This talk is based on a series of recent collaborative works together with Bartosz Malman (KTH).