Similar models of Toeplitz operators

D. V. Yakubovich

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 113-123

Résumé

We consider Toeplitz operators $T_F$ on Banach spaces $B(\Omega)$ satisfying some natural constraints, where $\Omega$ is a domain bounded by a simple piecewise smooth closed curve $\partial\Omega$. Assume that $F$ is a meromorphic function in a neighbourhood of $\bar\Omega$ and has no poles on $\partial\Omega$. Then $T_F$ is well-defined and bounded on $B(\Omega)$. It is proved that if the winding number of the curve $F|\partial\Omega$ with respect to every point $\lambda\in\mathbb C\setminus F(\partial\Omega)$ is nonnegative (and if some additional assumptions hold), then $T_F$ is similar to the multiplication by a function $\nu$ on some space $B_F(\sigma_*)$ of analytic functions on a Riemann surface $\sigma_*=\sigma_*(T_F)$; moreover, $\nu$ is nothing but the projection of $\sigma_*$ into $\mathbb C$. The surface $\sigma_*$ (which is called the ultraspectrum of $T_F$) and the Banach space $B_F(\sigma_*)$ are calculated explicitly and the equation $\nu(\sigma_*)=\operatorname{int}\sigma(T_F)$ holds.