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Distance estimate in the space of Toeplitz operators. (Russian) Zbl 0584.47024

Let \({\mathcal R}\) be an ultra-weakly closed linear subspace of the algebra L(H) of all linear and bounded operators in a Hilbert space H. Denote by \({\mathcal P}(H)\) the set of all orthogonal projections in L(H).
Let L(\({\mathcal R})=\{(Q,P)\in {\mathcal P}(H)\times {\mathcal P}(H)\), Q\({\mathcal R}P=0\}\) and denote by \(\ell ({\mathcal R})\) the set of its maximal elements with respect to natural order. In the case \({\mathcal R}={\mathcal R}_ 0\) consists of analytic Toeplitz operators in the Hardy space over the unit disc, the author proves the following formula:
For any bounded (essentially) function f on the unit circle \[ dist(T_ f,{\mathcal R}_ 0)=\sup \{\| QT_ fP\|,(Q,P)\in \ell ({\mathcal R}_ 0)\}, \] where \(T_ f\) is the Toeplitz operator corresponding to f.
As a consequence the norm of the derivation \(\Delta_{T_ f}:A\to AT_ f-T_ fA\) restricted to the algebra \(\tilde {\mathcal R}=\{T_{\phi},\phi\) runs over the algebra \(H^{\infty}\}\) also equals \(dist(T_ f,\tilde {\mathcal R})\).
Reviewer: J.Janas

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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