On the distance between unitary orbits of weighted shifts

Abstract:

In this paper, we consider invertible bilateral weighted shift operators acting on a complex separable Hilbert space $\mathcal {H}$. They have the property that there exist a constant $\tau > 0$ and an orthonormal basis ${\{ {{e_i}} \}_{i \in \mathbb {Z}}}$ for $\mathcal {H}$ with respect to which a shift $V$ acts by $W{e_i}= {w_i}{e_{i + 1}},i \in \mathbb {Z}$ and ${\mathbf {|}}{w_i}{\mathbf {|}} \geq \tau$. The equivalence class $\mathcal {U}(W)= \{ {U^{\ast }}\;WU:U \in \mathcal {B}(\mathcal {H}),U\;{\text {unitary}}\}$ of weighted shifts with weight sequence (with respect to the basis ${\{ {U^{\ast }}{e_i}\} _{i \in \mathbb {Z}}}$ for $\mathcal {H})$ identical to that of $W$ forms the unitary orbit of $W$. Given two shifts $W$ and $V$, one can define a distance $d(\mathcal {U}(V),\mathcal {U}(W))= \inf \{\parallel X - Y\parallel :X \in \mathcal {U}(V),Y \in \mathcal {U}(W)\}$ between the unitary orbits of $W$ and $V$. We establish numerical estimates for upper and lower bounds on this distance which depend upon information drawn from finite dimensional restrictions of these operators.