On operator-valued cosine sequences on UMD spaces

Wojciech Chojnacki
2010 Studia Mathematica  
A two-sided sequence (c n ) n∈Z with values in a complex unital Banach algebra is a cosine sequence if it satisfies c n+m + c n−m = 2c n c m for any n, m ∈ Z with c 0 equal to the unity of the algebra. A cosine sequence (c n ) n∈Z is bounded if sup n∈Z c n < ∞. A (bounded) group decomposition for a cosine sequence c = (c n ) n∈Z is a representation of c as c n = (b n + b −n )/2 for every n ∈ Z, where b is an invertible element of the algebra (satisfying sup n∈Z b n < ∞, respectively). It is
more » ... ctively). 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.
doi:10.4064/sm199-3-4 fatcat:s6c6ppoe7rahrndwrlcnjqq67a