D. Luecking has recently proved that a Toeplitz operator with measure symbol on the Bergman space of unit disk has finite rank if and only if its symbols is a linear combination of point masses. In this note we extend this theorem to higher dimensional cases.
Boo Rim CHOE. "On higher dimensional Luecking's theorem." J. Math. Soc. Japan 61 (1) 213 - 224, January, 2009. https://doi.org/10.2969/jmsj/06110213