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## On operator-valued cosine sequences on UMD spaces

### Tom 199 / 2010

Studia Mathematica 199 (2010), 267-278 MSC: 39B42, 47A60, 47D03, 47D09. DOI: 10.4064/sm199-3-4

#### Streszczenie

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.

#### Autorzy

• Wojciech ChojnackiSchool of Computer Science
and
Wydzia/l Matematyczno-Przyrodniczy
Szkoła Nauk Ścisłych
Uniwersytet Kardynała Stefana Wyszyńskiego
Dewajtis 5
01-815 Warszawa, Poland
e-mail

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