On the distance between unitary orbits of weighted shifts
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- by Laurent Marcoux PDF
- Trans. Amer. Math. Soc. 326 (1991), 585-612 Request permission
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.References
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Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 326 (1991), 585-612
- MSC: Primary 47B37; Secondary 47A30, 47C99
- DOI: https://doi.org/10.1090/S0002-9947-1991-1010887-9
- MathSciNet review: 1010887