On operator-valued cosine sequences on UMD spaces
Volume 199 / 2010
Abstract
A two-sided sequence $(c_n)_{n\in\mathbb{Z}}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m} + c_{n-m} = 2 c_n c_m$ for any $n,m \in \mathbb{Z}$ with $c_0$ equal to the unity of the algebra. A cosine sequence $(c_n)_{n\in\mathbb{Z}}$ is bounded if $\sup_{n \in \mathbb{Z}} \| c_n \| < \infty$. A (bounded) group decomposition for a cosine sequence $c = (c_n)_{n\in\mathbb{Z}}$ is a representation of $c$ as $c_n= (b^n + b^{-n})/2$ for every $n \in \mathbb{Z}$, where $b$ is an invertible element of the algebra (satisfying $\sup_{n \in \mathbb{Z}} \| b^n \| < \infty$, 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 $\mathcal{L}(X)$ denoting the algebra of all bounded linear operators on $X$, if $c$ is an $\mathcal{L}(X)$-valued bounded cosine sequence, then the standard group decomposition of $c$ is bounded.
