Georgia Institute of Technology College of Sciences

This is a two-part talk based on two completely separate projects. 

In the first part of this talk, we will discuss recent results on the structure of submodules in the Hardy space on the bidisk. 
The study of the structure of shift-invariant subspaces (or submodules) of the Hardy Space on the bidisk has been the subject of extensive research over the past several decades. Rudin's book, "Function Theory in Polydiscs", provides essential groundwork for the development of this area; however, still to this date we lack a complete characterization of the structure of the submodules of $H^2(\mathbb{D}^2)$ and there are not many concrete examples of inner functions in two variables representing each of the classes of inner functions detailed in his text. In 1994, Nakazi conjectured that all submodules whose shift cross commutator $[R_w, R_z^*]$ is self-adjoint are necessarily of Beurling type so that $rank[R_w, R_z^*] = 0$. A construction known as the Izuchi-Ohno type submodule establishes the existence of submodules with $[R_w, R_z^*] = [R_w, R_z^*]^*$ and $rank[R_w, R_z^*] = 1$. For $0 < |r|< 1$  there exists an inner function $\psi\in H^2(\mathbb{D}^2)$ such that  $$M = \psi \bigg(H^2(\mathbb{D}^2) \oplus \bigg( \underset{j \geq0} \bigoplus \, \mathbb{C} \cdot z^j \frac{\overline{w}}{1-rz\overline{w}} \bigg)\bigg)$$ is a submodule in $H^2(\mathbb{D}^2)$ of the Izuchi-Ohno type. The inner function's explicit construction and its properties are not given in their work; however, in this talk we will give the exact form of the inner function as well as show that all inner functions satisfying a particular property $\psi$ must also satisfy are examples of inner functions that are "not good". This is joint work with Rongwei Yang and Kelly Bickel. 
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In the second part of this talk, we will discuss recent results on continuous frames for Hilbert spaces generated by a $C_0$-semigroup. 
Due to Christensen, Hasannasab, and Philipp  we have a necessary and sufficient condition for a system $\{T^n \varphi\}_{n \in \mathbb{Z}_+}$ to be a frame, for a separable infinite-dimensional Hilbert space $H$, which exhibits the connection between frame theory and operator theory as the bounded operators $T$ that can be used to generate a frame for $H$ must be similar to a compression of the shift operator on $H^2(\mathbb{T})$ to a particular coinvariant subspace of the shift in $H^2(\mathbb{T})$. 
 Similarly, in recent work by Bailey, Han, Kornelson, Larson, and Liu it is shown that every frame representation for a countable, unital, left-cancellation semigroup $S$ must be equivalent to a compression of the left regular representation on $\ell^2(S)$ to a certain coinvariant subspace of the left regular representation in $\ell^2(S)$. 

When considering continuous frames given by a $C_0$-semigroup of operators $\{T(t)\}_{t \geq 0}$ we may ask whether a similar characterization for such frames can be obtained. Moreover, for a $C_0$-semigroup $\{T(t)\}_{t \geq 0}$, there is an associated bounded operator known as the cogenerator $C = (A+I)(A-I)^{-1}$ of the semigroup whenever $1 \notin \sigma(A)$ (where $A$ is the infinitesimal generator of the semigroup). It can be shown that $\{T(t)\varphi\}_{t \geq 0}$ is a continuous frame for $H$ if and only if $ \{C^n (A-I)^{-1}\varphi \}_{n \in \mathbb{Z}_+}$ is a discrete frame for $H$ so that studying the frame properties of the frames by iterations of the cogenerator will shed light on the frame properties of the continuous frames given by the associated semigroup $\{T(t)\}_{t \geq 0}$. In this talk, we will address the aforementioned questions as well as provide some results on frame properties of the frames obtained from iterations of operators defined in terms of the infinitesimal generator of a given $C_0$-semigroup. This is joint work with Eva Gallardo-Gutierrez and Jonathan Partington.