On operator-valued cosine sequences on UMD spaces

• Autorzy: Wojciech Chojnacki
• Języki publikacji: EN

Abstrakty

EN

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 𝓛(X) denoting the algebra of all bounded linear operators on X, if c is an 𝓛(X)-valued bounded cosine sequence, then the standard group decomposition of c is bounded.

Kategorie tematyczne

• 47D09: Operator sine and cosine functions and higher-order Cauchy problems
• 47D03: Groups and semigroups of linear operators
• 47A60: Functional calculus
• 39B42: Matrix and operator equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 199
• Numer: 3
• Strony: 267-278

Daty

wydano

2010

Twórcy

autor

Wojciech Chojnacki

• School of Computer Science, The University of Adelaide, Adelaide, SA 5005, Australia
• Wydział Matematyczno-Przyrodniczy, Szkoła Nauk Ścisłych, Uniwersytet Kardynała Stefana Wyszyńskiego, Dewajtis 5, 01-815 Warszawa, Poland

Identyfikatory

DOI

10.4064/sm199-3-4