#### Location

Cupples I Room 218

#### Start Date

7-21-2016 3:00 PM

#### End Date

21-7-2016 3:20 PM

#### Description

We consider systems of vectors of form $$\{A^nh_i:\;i\in I, n\geq 0 $$ where $\{h_i\}_{i\in I}$ is a countable (finite or infinite) system of vectors in a separable Hilbert space $ \mathcal{H} $ and $ A\in B(\mathcal{H}) $ is a bounded operator. We show that a system of that form can never be both complete and minimal, and find conditions that the operator $ A $ needs to satisfy for the system to be a frame or a complete Bessel system. We also investigate systems of form $$ \{\pi(g) h_i:\;i\in I, g\in \Gamma \} $$ where $\pi$ is a unitary representation on $ \mathcal{H} $ of a discrete group $\Gamma$ and extract information about the spectrum of the operators $ \pi(g) $ when the system is minimal or complete and Bessel. The initial motivation for considering these problems comes from what is called dynamical sampling problem.

Frames and Bessel systems generated by the iterative actions of operators

Cupples I Room 218

We consider systems of vectors of form $$\{A^nh_i:\;i\in I, n\geq 0 $$ where $\{h_i\}_{i\in I}$ is a countable (finite or infinite) system of vectors in a separable Hilbert space $ \mathcal{H} $ and $ A\in B(\mathcal{H}) $ is a bounded operator. We show that a system of that form can never be both complete and minimal, and find conditions that the operator $ A $ needs to satisfy for the system to be a frame or a complete Bessel system. We also investigate systems of form $$ \{\pi(g) h_i:\;i\in I, g\in \Gamma \} $$ where $\pi$ is a unitary representation on $ \mathcal{H} $ of a discrete group $\Gamma$ and extract information about the spectrum of the operators $ \pi(g) $ when the system is minimal or complete and Bessel. The initial motivation for considering these problems comes from what is called dynamical sampling problem.