THE COMMUTANT OF A CERTAIN COMPRESSION

Let G be any bounded region in the complex plane and K subset-of G be a simple compact arc of class C1. Let A2(G\K) (resp. A2(G)) be the Ber gman space on G\K (resp. G). Let S be the operator multiplication by o n A2(G\K) and C = P(N)S\N be the compression of S to the semi-invarian t subspace N = A2(G\K) - A2(G). We show that the commutant of C is th e set of all operators of the form A-1M(h)A, where h is a multiplier o n a certain Sobolev space of functions on K and (Af)(w) = integral(G) f(z)(zBAR - wBAR)-1 d A(z) (w is-an-element-of K) . We also use multip lier theory in fractional order Sobolev spaces to obtain further infor mation about C.