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 H \mathcal {H} . They have the property that there exist a constant τ > 0 \tau > 0 and an orthonormal basis { e i } i ∈ Z {\{ {{e_i}} \}_{i \in \mathbb {Z}}} for H \mathcal {H} with respect to which a shift V V acts by W e i = w i e i + 1 , i ∈ Z W{e_i}= {w_i}{e_{i + 1}},i \in \mathbb {Z} and | w i | ≥ τ {\mathbf {|}}{w_i}{\mathbf {|}} \geq \tau . The equivalence class U ( W ) = { U ∗ W U : U ∈ B ( H ) , U unitary } \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 ∗ e i } i ∈ Z {\{ {U^{\ast }}{e_i}\} _{i \in \mathbb {Z}}} for H ) \mathcal {H}) identical to that of W W forms the unitary orbit of W W . Given two shifts W W and V V , one can define a distance d ( U ( V ) , U ( W ) ) = inf { ∥ X − Y ∥: X ∈ U ( V ) , Y ∈ U ( W ) } 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 W and V 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.