On group decompositions of bounded cosine sequences

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.