On a class of jointly hyponormal Toeplitz operators

Abstract:

We characterize when a pair of Toeplitz operators $\mathbf {T}=(T_{\phi },T_{\psi })$ is jointly hyponormal under various assumptions—for example, $\phi$ is analytic or $\phi$ is a trigonometric polynomial or $\phi -\psi$ is analytic. A typical characterization states that $\mathbf {T}=(T_{\phi },T_{\psi })$ is jointly hyponormal if and only if an algebraic relation of $\phi$ and $\psi$ holds and the single Toeplitz operator $T_{\omega }$ is hyponormal, where $\omega$ is a combination of $\phi$ and $\psi$. More general results for an $n$-tuple of Toeplitz operators are also obtained.