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

 

A note on polynomial operator approximation


Authors: R. R. Smith and J. D. Ward
Journal: Proc. Amer. Math. Soc. 88 (1983), 491-494
MSC: Primary 47A30; Secondary 47A55
MathSciNet review: 699420
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Abstract: An example is given of an operator $ T$ contained in a block-diagonal algebra of operators $ \mathcal{A}$, an ideal $ J \subset \mathcal{A}$ and an infinite set of polynomials $ \mathcal{P}$ for which there is a $ K \in J$ satisfying $ {\left\Vert {p(T + K)} \right\Vert _{\mathcal{A}}} = {\left\Vert {p(T + K)} \right\Vert _{\mathcal{A}/J}}$ for any finite subset of $ \mathcal{P}$ but for which there is no $ K \in J$ satisfying $ {\left\Vert {p(T + K)} \right\Vert _{\mathcal{A}}} = {\left\Vert {p(T + K)} \right\Vert _{\mathcal{A}/J}}$ for all $ p \in \mathcal{P}$. This sheds some light on a well-known question of C. Olsen.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0699420-X
PII: S 0002-9939(1983)0699420-X
Keywords: Block diagonal operator
Article copyright: © Copyright 1983 American Mathematical Society