On operator-valued cosine sequences on UMD spaces
Date
2010
Authors
Chojnacki, W.
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Studia Mathematica, 2010; 199(3):267-278
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Wojciech Chojnacki
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Abstract
A two-sided sequence (c<inf>n</inf>)<inf>n∈Z</inf> with values in a complex unital Banach algebra is a cosine sequence if it satisfies c <inf>n+m</inf> + c<inf>n-m</inf> = 2c<inf>n</inf>c<inf>m</inf> for any n, m ∈ Z with C<inf>0</inf> equal to the unity of the algebra. A cosine sequence (c<inf>n</inf>)<inf>n∈Z</inf> is bounded if sup<inf>n∈Z</inf> ||c<inf>n</inf>|| < ∞. A (bounded) group decomposition for a cosine sequence c = (c<inf>n</inf>)<inf>n∈Z</inf> is a representation of c as C<inf>n</inf> = (b<sup>n</sup> + b<sup>-n</sup>)/2 for every n∈Z, where b is an invertible element of the algebra (satisfying sup<inf>n∈Z</inf> ||b<sup>n</sup>|| < ∞, respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, the so-called standard group decomposition. Here it is shown that if X is a complex UMD Banach space and, with ℒ (X) denoting the algebra of all bounded linear operators on X, if c is an ℒ (X)-valued bounded cosine sequence, then the standard group decomposition of c is bounded. © 2010 Instytut Matematyczny PAN.
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