TOEPLITZ OPERATORS AND THE ESSENTIAL BOUNDARY ON POLYANALYTIC FUNCTIONS

Let j be a nonzero integer and let U be a bounded domain. We construct a Fredholm symbol calculus for the C*-algebra generated by the poly-Bergman projection and the operators of multiplication by continuous functions. We define the j-removal boundary in Hilbert spaces of polyanalytic functions and prove that the quotient poly-Toeplitz C*-algebra generated by cosets of poly-Toeplitz operators with continuous symbols is *-isomorphic to the C*-algebra of continuous functions over the j-essential boundary. Unlike the Bergman case, we also show that if j ≠ ±1 then the j-essential boundary coincides with the set of non-isolated points on the boundary.