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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the distance between unitary orbits of weighted shifts


Author: Laurent Marcoux
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
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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{\vert}}{w_i}{\mathbf{\vert}} \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.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1010887-9
Article copyright: © Copyright 1991 American Mathematical Society

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