Linearly similar models of Toeplitz operators. (Russian. English summary) Zbl 0647.47041

Yakubovich, D. V.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 157, 113-123 (1987).

The problem of similarity of Toeplitz operators in Hardy spaces was investigated in many works. In this work the author considers Toeplitz operators \(T_ F\) on Banach spaces B(D) (including Hardy, Bergman, Bloch etc.) where D is a domain bounded by a simple piecewise smooth closed curve \(\partial D\). Namely for meromorphic F in a neighbourhood of \(\bar D,\) which has no poles on \(\partial D\), \(T_ F\) is bounded in B(D). In the case the winding number of the curve F(\(\partial D)\) with respect to every point \(z\in {\mathbb{C}}\setminus F(\partial D)\) is nonnegative (and if some additional conditions hold) the operator \(T_ F\) is similar to the multiplication by a function h on some space \(B_ F(\sigma_*)\) of analytic functions on a Riemann surface \(\sigma_*=\sigma_*(T_ F)\). The function h turns out to be the projection of \(\sigma_*\) into \({\mathbb{C}}\) the surface \(\sigma_*\) and the Banach space \(B_ F(\sigma_*)\) are computed in terms of F and B(D). This is a quite nice and clearly written paper.

Reviewer: J.Janas

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A45	Canonical models for contractions and nonselfadjoint linear operators

Keywords:

Toeplitz operators in Hardy spaces; winding number