Communications of the
Korean Mathematical Society CKMS

Suppose $T_\varphi$ is a $2$--hyponormal Toeplitz operator whose self--commutator has rank $n\ge 1$. If $H_{\bar\varphi}\left(\ker\,[T_\varphi^*, T_\varphi]\right)$ contains a vector $e_n$ in a canonical orthonormal basis $\{e_k\}_{k\in \mathbb Z_+}$ of $H^2(\mathbb T)$, then $\varphi$ should be an analytic function of the form $\varphi=q h$, where $q$ is a finite Blaschke product of degree at most $n$ and $h$ is an outer function.

Keywords: Toeplitz operators, finite rank self-commutators, subnormal, hyponormal, 2--hyponormal

MSC numbers: Primary 47B20, 47B35