On operator-valued cosine sequences on UMD spaces

Wojciech Chojnacki¹

¹ School of Computer Science The University of Adelaide Adelaide, SA 5005, Australia and Wydzia/l Matematyczno-Przyrodniczy Szkoła Nauk Ścisłych Uniwersytet Kardynała Stefana Wyszyńskiego Dewajtis 5 01-815 Warszawa, Poland

Studia Mathematica, Tome 199 (2010) no. 3, pp. 267-278

Résumé

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.

Détail

DOI : 10.4064/sm199-3-4

Keywords: two sided sequence mathbb values complex unital banach algebra cosine sequence satisfies n m mathbb equal unity algebra cosine sequence mathbb bounded sup mathbb infty bounded group decomposition cosine sequence mathbb representation n every mathbb where invertible element algebra satisfying sup mathbb infty respectively known every bounded cosine sequence possesses universally defined group decomposition so called standard group decomposition here shown complex umd banach space mathcal denoting algebra bounded linear operators mathcal valued bounded cosine sequence standard group decomposition bounded