On operator-valued cosine sequences on UMD spaces

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 supn∈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, 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.