Toeplitz Operators with Homogeneous Symbols on Polyharmonic Spaces

We describe C*-algebras generated by Toeplitz operators with homogeneous symbols acting on polyharmonic Bergman spaces of the upper half-plane Pi. The symbols considered here have finite limits at the points 0 and pi. Under these conditions on the family of symbols, a Toeplitz operator acting on the true polyharmonic space H-(n)(2)(Pi) is unitarily equivalent to a 2x2 matrix-valued function defined on (R) over bar. The C*-algebra generated by these matrix-valued functions turns out to be isomorphic to the algebra C := {f = (fij) is an element of M-2(C (R) over bar)) : f (+/-infinity) is diagonal, f(11)(+/-infinity) = f(22)(-/+infinity)}. Besides, we prove that the C*-algebra generated by Toeplitz operators with homogeneous symbols, acting on the polyharmonic Bergman space H-n(2)(Pi), is isomorphic to the C*-subalgebra of M-2n(C((R) over bar)) consisting of all matrix-valued functions f = (fij) such that f(-infinity) = (lambda I-1 0I 0I lambda I-2), f(+infinity) = (lambda I-2 0I 0I lambda I-1), lambda(1), lambda(2) is an element of C, where I is the n x n identity matrix.