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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Best approximation and quasitriangular algebras
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by Timothy G. Feeman PDF
Trans. Amer. Math. Soc. 288 (1985), 179-187 Request permission

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 $\left \| {T - A} \right \| = \inf \{ \left \| {T - S} \right \|:S \in \operatorname {Alg} \mathcal {P} + \mathcal {K}\}$? 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
  • © Copyright 1985 American Mathematical Society
  • 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