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Type:  Journal article 
Title:  On group decompositions of bounded cosine sequences 
Author:  Chojnacki, W. 
Citation:  Studia Mathematica, 2007; 181(1):6185 
Publisher:  Polish Acad Sciences Inst Mathematics 
Issue Date:  2007 
ISSN:  00393223 17306337 
Statement of Responsibility:  Wojciech Chojnacki 
Abstract:  A twosided sequence (cn)n∈Z with values in a complex unital Banach algebra is a cosine sequence if it satisfies c n+m + cnm = 2cncm for any n, m ∈ ℤ with c0 equal to the unity of the algebra. A cosine sequence (cn)n∈Z is bounded if sup n∈Zcn < ∞. A (bounded) group decomposition for a cosine sequence c = (cn)n∈Z is a representation of c as cn = (bn + bn)/2 for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying supn∈Z bn < ∞, respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, here referred to as a standard group decomposition. The present paper reveals various classes of bounded operatorvalued cosine sequences for which the standard group decomposition is bounded. One such class consists of all bounded ℒ(X)valued cosine sequences (cn)n∈Z, with X a complex Banach space and ℒ(X) the algebra of all bounded linear operators on X, for which c 1 is scalartype prespectral. Every bounded ℒ(H)valued cosine sequence, where H is a complex Hilbert space, falls into this class. A different class of bounded cosine sequences with bounded standard group decomposition is formed by certain ℒ(X)valued cosine sequences (cn) n∈Z, with X a reflexive Banach space, for which c1 is not scalartype spectralin fact, not even spectral. The isolation of this class uncovers a novel family of nonprespectral operators. Examples are also given of bounded ℒ(H)valued cosine sequences, with H a complex Hilbert space, that admit an unbounded group decomposition, this being different from the standard group decomposition which in this case is necessarily bounded. © Instytut Matematyczny PAN, 2007. 
DOI:  10.4064/sm18115 
Published version:  http://journals.impan.gov.pl/sm/Inf/18115.html 
Appears in Collections:  Aurora harvest 6 Computer Science publications 
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