Toeplitz operators on Cartan domains essentially commute with a bilateral shift

Abstract:

For bounded symmetric domains $\Omega \subset {{\mathbf {C}}^N}$, a bilaterial shift operator $U$ is shown to exist on the Bergman space ${A^2}(\Omega )$ such that $U{T_f} - {T_f}U$ is a compact operator for all Toeplitz operators ${T_f}$. This may be viewed as an extension of the well-known fact that ${S^{\ast }}TS - T = 0$ whenever $T$ is a Toeplitz operator on ${H^2},\;S$ being the unilateral shift. It also follows that the ${C^{\ast }}$-algebra generated by Toeplitz operators on ${A^2}(\Omega )$ does not contain all bounded operators.