Algebraic properties and the finite rank problem for Toeplitz operators on the Segal–Bargmann space

We study three different problems in the area of Toeplitz operators on the Segal–Bargmann space in Cn. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class Sym>0(Cn) of symbols having certain growth at infinity. We then provide explicit examples of zero-products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal–Bargmann space and on the Bergman space over the unit ball. Finally, we discuss the “finite rank problem”. We show that there are no non-trivial rank one Toeplitz operators Tf for f∈Sym>0(Cn). In all these problems, the growth at infinity of the symbols plays a crucial role.