On group decompositions of bounded cosine sequences

Wojciech Chojnacki. "On group decompositions of bounded cosine sequences." Studia Mathematica 181.1 (2007): 61-85. <http://eudml.org/doc/284941>.

@article{WojciechChojnacki2007,
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, 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 ℒ(X)-valued cosine sequences $(cₙ)_\{n∈ℤ\}$, with X a complex Banach space and ℒ(X) the algebra of all bounded linear operators on X, for which c₁ is scalar-type prespectral. Every bounded 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 ℒ(X)-valued cosine sequences $(cₙ)_\{n∈ℤ\}$, with X a reflexive Banach space, for which c₁ 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 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.},
author = {Wojciech Chojnacki},
journal = {Studia Mathematica},
keywords = {cosine sequence; cosine function; group decomposition; scalar-type prespectral operator; doubly power bounded operator},
language = {eng},
number = {1},
pages = {61-85},
title = {On group decompositions of bounded cosine sequences},
url = {http://eudml.org/doc/284941},
volume = {181},
year = {2007},
}

TY - JOUR
AU - Wojciech Chojnacki
TI - On group decompositions of bounded cosine sequences
JO - Studia Mathematica
PY - 2007
VL - 181
IS - 1
SP - 61
EP - 85
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, 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 ℒ(X)-valued cosine sequences $(cₙ)_{n∈ℤ}$, with X a complex Banach space and ℒ(X) the algebra of all bounded linear operators on X, for which c₁ is scalar-type prespectral. Every bounded 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 ℒ(X)-valued cosine sequences $(cₙ)_{n∈ℤ}$, with X a reflexive Banach space, for which c₁ 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 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.
LA - eng
KW - cosine sequence; cosine function; group decomposition; scalar-type prespectral operator; doubly power bounded operator
UR - http://eudml.org/doc/284941
ER -