The Commutant of a Certain Compression

Authors

William T. Ross, University of RichmondFollow

DOI

10.1090/S0002-9939-1993-1145951-4

Abstract

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 = P_N 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 M_hA , 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.

Document Type

Article

Publication Date

1993

Publisher Statement

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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Recommended Citation

Ross, William T. "The Commutant of a Certain Compression." Proceedings of the American Mathematical Society 118, no. 3 (1993): 831-37. doi:10.1090/S0002-9939-1993-1145951-4.