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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Best approximation and quasitriangular algebras


Author: Timothy G. Feeman
Journal: Trans. Amer. Math. Soc. 288 (1985), 179-187
MSC: Primary 47D25; Secondary 41A35, 41A65, 47A66
DOI: https://doi.org/10.1090/S0002-9947-1985-0773055-9
MathSciNet review: 773055
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Abstract: If $ \mathcal{P}$ is a linearly ordered set of projections on a Hilbert space and $ \mathcal{K}$ is the ideal of compact operators, then $ \operatorname{Alg}\, \mathcal{P} + \mathcal{K}$ is the quasitriangular algebra associated with $ \mathcal{P}$. We study the problem of finding best approximants in a given quasitriangular algebra to a given operator: given $ T$ and $ \mathcal{P}$, is there an $ A$ in $ \operatorname{Alg}\, \mathcal{P} + \mathcal{K}$ such that $ \left\Vert {T - A} \right\Vert = \inf \{ \left\Vert {T - S} \right\Vert:S \in \operatorname{Alg}\,\mathcal{P} + \mathcal{K}\} $? We prove that if $ \mathcal{A}$ is an operator subalgebra which is closed in the weak operator topology and satisfies a certain condition $ \Delta $, then every operator $ T$ has a best approximant in $ \mathcal{A} + \mathcal{K}$. We also show that if $ \mathcal{E}$ is an increasing sequence of finite rank projections converging strongly to the identity then $ \operatorname{Alg}\,\mathcal{E}$ satisfies the condition $ \Delta $. Also, we show that if $ T$ is not in $ \operatorname{Alg}\,\mathcal{E} + \mathcal{K}$ then the best approximants in $ \operatorname{Alg}\,\mathcal{E} + \mathcal{K}$ to $ T$ are never unique.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0773055-9
Keywords: Quasitriangular operator algebras, nest algebra, best approximation
Article copyright: © Copyright 1985 American Mathematical Society

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