Invariant subspaces of vector Hardy spaces of a multiply connected domain
by
Dmitry Yakubovich
Departamento de Matemáticas (Universidad Autónoma de Madrid)
Coauthors: Alexander Kiselev (Saint Petersburg State University)

Let R be a a multiply connected domain. We discuss the problem of describing all invariant subspaces of the operator of multiplication by the independent variable on the Hardy space H^p(R), 1 ≤ p < ∞.

If R is a simply connected domain, then, due to the Beurling-Srinivasan Theorem, all invariant subspaces E of H^p(R) have the form E=qH^p(R), where q is an inner function in R. The description of invariant subspaces of H² of a multiply connected domain was obtained by Hitt and Sarason in 1988 for the case of the annulus and by the speaker in 1989 for more general domains R and for general values of p. These descriptions involve the so-called nearly invariant subspaces M of H² of the unit disc, which are characterized by the following property:

If f is in M and f(0)=0, then [f(z)/z] is also in M.

  Nearly invariant subspaces appear in several contexts; for instance, the kernels of classical Toeplitz operators are always nearly invariant (but not vise versa).

In this talk, we present a description of invariant subspaces of the vector Hardy space H²(R, Cⁿ), which we obtained recently. We will also mention the results by V. Kapustin, which relate this topic with functional models of non-dissipative operators.

Date received: June 13, 2008