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On the distance between unitary orbits of weighted shifts

In this paper, we consider invertible bilateral weighted shift operators acting on a complex separable Hilbert space X. They have the property that there exist a constant T > 0 and an orthonormal basis {ei}iEZ for X with respect to which a shift V acts by Wei = wie1+1, i E Z and I wi I > T . The equivalence class ?/(W) = {U* WU: U E (), U unitary} of weighted shifts with weight sequence (with respect to the basis {U*e IiEZ for X) identical to that of W forms the unitary orbit of W. Given two shifts W and V, one can define a distance d(W (V), W'(W)) = inf{lIX Yl: X e /(V), Y E 91(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.