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An operator version of Abel's continuity theorem
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Vaja Tarieladze
Published/Copyright:
November 15, 2010
Abstract
For a Banach space X let π be the set of continuous linear operators A : X β X with βAβ < 1, I be the identity operator and
πc β {A β π : βI β Aβ β€ c(1 β βAβ)},
where c β₯ 1 is a constant. Let, moreover, (xk)kβ₯0 be a sequence in X such that the series 
converges and Ζ : π βͺ {I} β X be the mapping defined by the equality
It is shown that Ζ is continuous on π and for every c β₯ 1 the restriction of Ζ to πc βͺ {I} is continuous at I.
Received: 2009-06-16
Published Online: 2010-11-15
Published in Print: 2010-December
Β© de Gruyter 2010
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Articles in the same Issue
- On a problem concerning quaternion valued Gaussian random variables
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- A limit theorem on symmetric matrices
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- Proof of the Zalcman conjecture for initial coefficients
- Recursive parameter estimation in the trend coefficient of a diffusion process
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- Asymptotic efficiency of exponentiality tests based on order statistics characterization
- Universal truncation error upper bounds in sampling restoration
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