The UCF Mathematics Colloquium, held every Monday from 3:30pm to 4:30pm in MSB 318, offers a diverse platform for research scholars, faculty, students, and industry experts to share and exchange ideas, fostering discussion and networking across various areas of mathematics.
Dr. Victor Bailey from University of Oklahoma will speak at this week's colloquium on characterizing submodules in H^2(D^2) using the core function.
Abstract: It is well known that $H^2(\mathbb{D}^2)$ is a RKHS with the reproducing kernel
$K(\lambda,z) = \frac{1}{(1-\overline{\lambda_1}z_1)(1-\overline{\lambda_2}z_2)}$ and that for any submodule $M \subseteq H^2(\mathbb{D}^2)$ its reproducing kernel is $K^M (\lambda, z) = P_M K(\lambda, z)$ where $P_M$ is the orthogonal projection onto $M$. Associated with any submodule $M$ are the core function $G^M (\lambda, z) = \frac{K^M (\lambda,z)}{K(\lambda,z)}$ and the core operator $C_M$, an integral transform on $H^2(\mathbb{D}^2)$ with kernel function $G^M$. 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.
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