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Toeplitz operators on certain weakly pseudoconvex domains

Part of: Selfadjoint operator algebras Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

David Crocker  and
Iain Raeburn
David Crocker
Affiliation:
School of Mathematics, University of New South Wales, Post Office Box 1, Kensington, N.S.W. 2033, Australia
Iain Raeburn
Affiliation:
School of Mathematics, University of New South Wales, Post Office Box 1, Kensington, N.S.W. 2033, Australia
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Abstract

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Let Ω be the weakly pseudoconvex domain

and let ∂Ω be its boundary. If ϕ ∈ L∞ (∂Ω), we denote by Tϕ, the Toephtz operator with symbol ϕ acting on the Hardy space H2(∂Ω), and by J(∂Ω) the C*-subalgebra of B(H2(∂Ω)) generated by the Toeplitz operators with continuous symbol. Our main theorem asserts that J(∂Ω) contains the ideal K of all compact operators on H2(∂Ω), and that the symbol map ϕ→Tϕ induces an isomorphism of C(∂Ω) onto the quotient C*-algebra ℑ(∂Ω)/K. Similar results have been established before for other domains, and in particular when Ω is strongly pseudoconvex. The main interest of our results lies in their proofs: ours are elementary, whereas those used in the strongly pseudoconvex case depend heavily on the theory of the tangential Cauchy-Riemann operator.

MSC classification

Secondary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 1 , June 1981 , pp. 1 - 14
Copyright
Copyright © Australian Mathematical Society 1981

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