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

Toeplitz spectral inclusion and generalized in modulus property


Author: J. Janas
Journal: Proc. Amer. Math. Soc. 104 (1988), 231-234
MSC: Primary 47B35
MathSciNet review: 958073
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Abstract: Let $ A \subset {C_b}(X)$ be a function algebra and let $ \mu $ be a Borel measure on $ X$ and $ {H^2}(\mu ) = \bar A$--the closure in $ {L^2}(\mu )$. It turns out that the spectral inclusion theorem for Toeplitz operators, defined in the above context, implies the density of finite sums $ \sum\nolimits_i {{{\left\vert {{u_i}} \right\vert}^2},{u_i} \in A} $, in

(i) the cone of positive functions in $ {L^1}(\mu )$,

(ii) the cone of positive functions in $ {L^p}(\mu ),p \leq 2$, if $ \mu (X){\text{ < }}\infty $,

(iii) the cone of positive functions in $ C(X)$, if $ X$ is compact.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0958073-8
PII: S 0002-9939(1988)0958073-8
Article copyright: © Copyright 1988 American Mathematical Society