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https://hdl.handle.net/2440/42014
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dc.contributor.author | Chojnacki, W. | - |
dc.date.issued | 2007 | - |
dc.identifier.citation | Studia Mathematica, 2007; 181(1):61-85 | - |
dc.identifier.issn | 0039-3223 | - |
dc.identifier.issn | 1730-6337 | - |
dc.identifier.uri | http://hdl.handle.net/2440/42014 | - |
dc.description.abstract | A two-sided sequence (cn)n∈Z with values in a complex unital Banach algebra is a cosine sequence if it satisfies c n+m + cn-m = 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∈Z||cn|| < ∞. A (bounded) group decomposition for a cosine sequence c = (cn)n∈Z is a representation of c as cn = (bn + b-n)/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 operator-valued 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 scalar-type 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 scalar-type spectral-in fact, not even spectral. The isolation of this class uncovers a novel family of non-prespectral 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. | - |
dc.description.statementofresponsibility | Wojciech Chojnacki | - |
dc.language.iso | en | - |
dc.publisher | Polish Acad Sciences Inst Mathematics | - |
dc.source.uri | http://journals.impan.gov.pl/sm/Inf/181-1-5.html | - |
dc.title | On group decompositions of bounded cosine sequences | - |
dc.type | Journal article | - |
dc.identifier.doi | 10.4064/sm181-1-5 | - |
pubs.publication-status | Published | - |
dc.identifier.orcid | Chojnacki, W. [0000-0001-7782-1956] | - |
Appears in Collections: | Aurora harvest 6 Computer Science publications |
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