On operator-valued cosine sequences on UMD spaces

A two-sided sequence (c(n))(n is an element of z) with values in a complex unital Banach algebra is a cosine sequence if it satisfies c(n+m) + (cn-m) = 2c(n)c(m) for any n, m is an element of Z with c(o) equal to the unity of the algebra. A cosine sequence (c(n))(nEz) is bounded if sup(n is an element of z) vertical bar vertical bar c(n)vertical bar vertical bar < infinity. A (bounded) group decomposition for a cosine sequence c = (c(n))(n is an element of z) is a representation of c as c(n) = (b(n) + b(-n))/2 for every n is an element of Z, where b is an invertible element of the algebra (satisfying sup(n is an element of z) vertical bar vertical bar b(n)vertical bar vertical bar < infinity, 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.