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2007 | 181 | 1 | 61-85
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On group decompositions of bounded cosine sequences

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A two-sided sequence $(cₙ)_{n∈ℤ}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m} + c_{n-m} = 2cₙcₘ$ 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 $sup_{n ∈ ℤ} ||bⁿ|| < ∞$, 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 c₁ 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.
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  • School of Computer Science, The University of Adelaide, Adelaide, SA 5005, Australia
  • Wydział Matematyczno-Przyrodniczy, Szkoła Nauk Ścisłych, Uniwersytet Kardynała Stefana Wyszyńskiego, Dewajtis 5, 01-815 Warszawa, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm181-1-5
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