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

作者：Bauer Wolfram^*; Le Trieu

来源：Journal of Functional Analysis, 2011, 261(9): 2617-2640.

DOI：10.1016/j.jfa.2011.07.006

• Commuting operators
• Finite rank problem
• Zero-products of Toeplitz operators

We study three different problems in the area of Toeplitz operators on the Segal-Bargmann space in C(n). 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)(C(n)) 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 T(f) for f is an element of Sym(>0)(C(n)). In all these problems, the growth at infinity of the symbols plays a crucial role.

• 出版日期2011-11-1