Robertson-type Theorems for Countable Groups of Unitary Operators

Abstract

Let $\mathcal{G}$ be a countably infinite group of unitary operators on a complex separable Hilbert space $H$. Let $X = \{x_{1},...,x_{r}\}$ and $Y = \{y_{1},...,y_{s}\}$ be finite subsets of $H$, $r < s$, $V_{0} = \bar{span} \mathcal{G}(X), V_1 = \bar{span} \mathcal{G}(Y)$ and $ V_{0} \subset V_{1} $. We prove the following result: Let $W_0$ be a closed linear subspace of $V_1$ such that $V_0 \oplus W_0 = V_1$ (i.e., $V_0 + W_0 = V_1$ and $V_0 \cap W_0 = \{0 \}$). Suppose that $\mathcal{G}(X)$ and $\mathcal{G}(Y)$ are Riesz bases for $V_{0}$ and $V_{1}$ respectively. Then there exists a subset $\Gamma =\{z_1,..., z_{s-r}\}$ of $W_0$ such that $\mathcal{G}(\Gamma)$ is a Riesz basis for $W_0$ if and only if $ g(W_0) \subseteq W_0 $ for every $g$ in $\mathcal{G}$. We first handle the case where the group is abelian and then use a cancellation theorem of Dixmier to adapt this to the non-abelian case. Corresponding results for the frame case and the biorthogonal case are also obtained.