On a class of jointly hyponormal Toeplitz operators

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.