On group decompositions of bounded cosine sequences

Wojciech Chojnacki¹

¹ School of Computer Science The University of Adelaide Adelaide, SA 5005 Australia and Wydzia/l Matematyczno-Przyrodniczy Szko/la Nauk /Scis/lych Uniwersytet Kardyna/la Stefana Wyszy/nskiego Dewajtis 5 01-815 Warszawa, Poland

Studia Mathematica, Tome 181 (2007) no. 1, pp. 61-85

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, 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.

Détail

DOI : 10.4064/sm181-1-5

Keywords: two sided sequence mathbb values complex unital banach algebra cosine sequence satisfies n m 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 here referred standard group decomposition present paper reveals various classes bounded operator valued cosine sequences which standard group decomposition bounded class consists bounded mathscr valued cosine sequences mathbb complex banach space mathscr algebra bounded linear operators which scalar type prespectral every bounded mathscr valued cosine sequence where complex hilbert space falls class different class bounded cosine sequences bounded standard group decomposition formed certain mathscr valued cosine sequences mathbb reflexive banach space which scalar type spectral even spectral isolation class uncovers novel family non prespectral operators examples given bounded mathscr valued cosine sequences complex hilbert space admit unbounded group decomposition being different standard group decomposition which necessarily bounded