On operator-valued cosine sequences on UMD spaces

Wojciech Chojnacki. "On operator-valued cosine sequences on UMD spaces." Studia Mathematica 199.3 (2010): 267-278. <http://eudml.org/doc/285832>.

@article{WojciechChojnacki2010,
abstract = {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.},
author = {Wojciech Chojnacki},
journal = {Studia Mathematica},
keywords = {cosine sequence; cosine function; group decomposition; UMD space; transference method},
language = {eng},
number = {3},
pages = {267-278},
title = {On operator-valued cosine sequences on UMD spaces},
url = {http://eudml.org/doc/285832},
volume = {199},
year = {2010},
}

TY - JOUR
AU - Wojciech Chojnacki
TI - On operator-valued cosine sequences on UMD spaces
JO - Studia Mathematica
PY - 2010
VL - 199
IS - 3
SP - 267
EP - 278
AB - 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.
LA - eng
KW - cosine sequence; cosine function; group decomposition; UMD space; transference method
UR - http://eudml.org/doc/285832
ER -