Density of algebras generated by Toeplitz operators on Bergman spaces

Abstract

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, the C^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spaces A²(Ω) for a wide class of plane domains Ω⊂C, and in Fock spaces A²(C^N), N≧1.