On group decompositions of bounded cosine sequences
Volume 181 / 2007
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, here referred to as a standard group decomposition. The present paper reveals various classes of bounded operator-valued cosine sequences for which the standard group decomposition is bounded. One such class consists of all bounded $\mathscr{L}(X)$-valued cosine sequences $(c_n)_{n\in\mathbb{Z}}$, with $X$ a complex Banach space and $\mathscr{L}(X)$ the algebra of all bounded linear operators on $X$, for which $c_1$ is scalar-type prespectral. Every bounded $\mathscr{L}(H)$-valued cosine sequence, where $H$ is a complex Hilbert space, falls into this class. A different class of bounded cosine sequences with bounded standard group decomposition is formed by certain $\mathscr{L}(X)$-valued cosine sequences $(c_n)_{n\in\mathbb{Z}}$, with $X$ a reflexive Banach space, for which $c_1$ is not scalar-type spectral—in fact, not even spectral. The isolation of this class uncovers a novel family of non-prespectral operators. Examples are also given of bounded $\mathscr{L}(H)$-valued cosine sequences, with $H$ a complex Hilbert space, that admit an unbounded group decomposition, this being different from the standard group decomposition which in this case is necessarily bounded.
