On Operator-valued Cosine Sequences on Umd Spaces

A two-sided sequence (cn)n∈Z with values in a complex unital Banach algebra is a cosine sequence if it satisfies cn+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 = (b n +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.