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 $H$, $r<s$, $V_0=\overline{span}\mathcal{G}(X),V_1=\overline{span}\mathcal{G}(Y)$ 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.