Let G be any bounded region in the complex plane and K Ϲ G be a simple compact arc of class C1. Let A2(G\K) (resp. A2(G)) be the Bergman space on G\K (resp. G). Let S be the operator multiplication by z on A2(G\K) and C = PN S│N be the compression of S to the semi-invariant subspace N = A2(G\K) Ɵ A2(G). We show that the commutant of C* is the set of all operators of the form A-1 MhA , where h is a multiplier on a certain Sobolev space of functions on K and (Af)(w) = ∫Gf(z)(z- - w-)-1 dA(z) (w ϵ K). We also use multiplier theory in fractional order Sobolev spaces to obtain further information about C.

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Copyright © 1993 American Mathematical Society. This article first appeared in Proceedings of the American Mathematical Society 118:3 (1993), 831-837.

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