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Proceedings of the American Mathematical Society

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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\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.


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

  • [1] C. A. Akemann and G. K. Pedersen, Ideal perturbations of elements in $ {C^ * }$-algebras, Math. Scand. 41 (1977), 117-139. MR 0473848 (57:13507)
  • [2] W. F. Donoghue, On the numerical range of a bounded operator, Michigan Math. J. 4 (1957), 261-263. MR 0096127 (20:2622)
  • [3] C. L. Olsen, Norms of compact perturbations of operators, Pacific J. Math. 68 (1977), 209-228. MR 0451010 (56:9300)
  • [4] C. L. Olsen and J. K. Plastiras, Quasialgebraic operators, compact perturbations and the essential norm, Michigan Math. J. 21 (1974), 385-397. MR 0365205 (51:1458)
  • [5] R. R. Smith and J. D. Ward, Locally isometric liftings from quotient $ {C^ * }$-algebras, Duke Math. J. 47 (1980), 621-631. MR 587170 (82a:46068)
  • [6] J. G. Stampfli and J. P. Williams, Growth conditions and the numerical range in a Banach algebra, Tôhoku Math. J. 20 (1968), 417-424. MR 0243352 (39:4674)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0699420-X
Keywords: Block diagonal operator
Article copyright: © Copyright 1983 American Mathematical Society

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