On operator-valued cosine sequences on UMD spaces

Wojciech Chojnacki

Studia Mathematica, Tome 196 (2010), p. 267-278 / Harvested from The Polish Digital Mathematics Library

Résumé

A two-sided sequence $(cₙ)_{n∈ℤ}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m} + c_{n-m} = 2cₙcₘ$ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence $(cₙ)_{n∈ℤ}$ is bounded if $sup_{n∈ℤ} ||cₙ|| < ∞$. A (bounded) group decomposition for a cosine sequence $c = (cₙ)_{n∈ℤ}$ is a representation of c as $cₙ = (bⁿ+b^{-n})/2$ for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying $sup_{n∈ℤ} ||bⁿ|| < ∞$, 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 (X) denoting the algebra of all bounded linear operators on X, if c is an (X)-valued bounded cosine sequence, then the standard group decomposition of c is bounded.

Détail

Publié le : 2010-01-01

Zbl 1215.47036

EUDML-ID : urn:eudml:doc:285832