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
DOI:
https://doi.org/10.1090/S0002-9939-1983-0699420-X
MathSciNet review:
699420
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Abstract | References | Similar Articles | Additional Information
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 ${\left \| {p(T + K)} \right \|_{\mathcal {A}}} = {\left \| {p(T + K)} \right \|_{\mathcal {A}/J}}$ for any finite subset of $\mathcal {P}$ but for which there is no $K \in J$ satisfying ${\left \| {p(T + K)} \right \|_{\mathcal {A}}} = {\left \| {p(T + K)} \right \|_{\mathcal {A}/J}}$ for all $p \in \mathcal {P}$. This sheds some light on a well-known question of C. Olsen.
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Additional Information
Keywords:
Block diagonal operator
Article copyright:
© Copyright 1983
American Mathematical Society