Nearly Invariant Spaces of Analytic Functions

Nearly Invariant Spaces of Analytic Functions

Fri, 27/02/2009
15:15 		Alexandru Aleman (NBFAS Meeting) (Lund) Functional Analysis Seminar Add to calendar L3
Nearly Invariant Spaces of Analytic Functions
We consider Hilbert spaces $ H $ which consist of analytic functions in a domain $ \Omega\subset\mathbb{C} $ and have the property that any zero of an element of $ H $ which is not a common zero of the whole space, can be divided out without leaving $ H $. This property is called near invariance and is related to a number of interesting problems that connect complex analysis and operator theory. The concept probably appeared first in L. de Branges' work on Hilbert spaces of entire functions and played later a decisive role in the description of invariant subspaces of the shift operator on Hardy spaces over multiply connected domains. There are a number of structure theorems for nearly invariant spaces obtained by de Branges, Hitt and Sarason, and more recently by Feldman, Ross and myself, but the emphasis of my talk will be on some applications; the study of differentiation invariant subspaces of $ C^\infty(\mathbb{R}) $, or invariant subspaces of Volterra operators on spaces of power series on the unit disc. Finally, we discuss near invariance in the vector-valued case and show how it can be related to kernels of products of Toeplitz operators. More precisely, I will present in more detail the solution of the following problem: If a finite product of Toeplitz operators is the zero operator then one of the factors is zero.