Hardy space
Let "S" denote the unit sphere in C"n" and define the Hardy space H2("S") be the closure in "L"2("S") of the restriction of polynomials in the coordinates "z"1, ..., "zn". Let "P" be the projection onto Hardy space. It is known that if "m"("f") denotes multiplication by a continuous function "f" on "S", then the commutator [P,"m"("f")] is compact. Consequently, defining the Toeplitz operator by