Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


The commutant of a certain compression

Author: William T. Ross
Journal: Proc. Amer. Math. Soc. 118 (1993), 831-837
MSC: Primary 47B38; Secondary 47A20, 47B35
MathSciNet review: 1145951
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be any bounded region in the complex plane and $ K \subset G$ be a simple compact arc of class $ {C^1}$. Let $ {A^2}(G\backslash K)$ (resp. $ {A^2}(G)$) be the Bergman space on $ G\backslash K$ (resp. $ G$). Let $ S$ be the operator multiplication by $ z$ on $ {A^2}(G\backslash K)$ and $ C = {P_\mathcal{N}}S{\vert _\mathcal{N}}$ be the compression of $ S$ to the semi-invariant subspace $ \mathcal{N} = {A^2}(G\backslash K) \ominus {A^2}(G)$. We show that the commutant of $ {C^{\ast}}$ is the set of all operators of the form $ {A^{ - 1}}{M_h}A$, where $ h$ is a multiplier on a certain Sobolev space of functions on $ K$ and $ (Af)(w) = \int_G {f(z){{(\overline z - \overline w )}^{ - 1}}dA(z)(w \in K)} $. We also use multiplier theory in fractional order Sobolev spaces to obtain further information about $ C$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B38, 47A20, 47B35

Retrieve articles in all journals with MSC: 47B38, 47A20, 47B35

Additional Information

PII: S 0002-9939(1993)1145951-4
Keywords: Bergman spaces, multiplication operators, Sobolev spaces, multipliers
Article copyright: © Copyright 1993 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia