Finite rank Toeplitz operators on the Bergman space

Abstract:

Given a complex Borel measure $\mu$ with compact support in the complex plane $\mathbb {C}$ the sesquilinear form defined on analytic polynomials $f$ and $g$ by $B_\mu (f,g) = \int f\bar g d\mu$, determines an operator $T_\mu$ from the space of such polynomials $\mathcal {P}$ to the space of linear functionals on $\overline {\mathcal {P}}$. This operator is called the Toeplitz operator with symbol $\mu$. We show that $T_\mu$ has finite rank if and only if $\mu$ is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.