Characterizing Submodules in $H^2(\mathbb{D}^2)$ Using the Core Function

Series
Time
Wednesday, January 29, 2025 - 2:00pm for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Victor Bailey – University of Oklahoma – victor.bailey@ou.edu
Organizer
Michael Lacey

It is well known that  H2(D2) is a RKHS with the reproducing kernel K(λ,z)=1(1λ1¯z1)(1λ2¯z2) and that for any submodule MH2(D2) its reproducing kernel is KM(λ,z)=PMK(λ,z) where PM is the orthogonal projection onto M. Associated with any submodule M are the core function GM(λ,z)=KM(λ,z)K(λ,z) and the core operator CM, an integral transform on H2(D2) with kernel function GM. The utility of these constructions for better understanding the structure of a given submodule is evident from the various works in the past 20 years. In this talk, we will discuss the relationship between the rank, codimension, etc. of a given submodule and the properties of its core function and core operator. In particular, we will discuss the longstanding open question regarding whether we can characterize all submodules whose core function is bounded. This is a joint project with Rongwei Yang.