On group decompositions of bounded cosine sequences

• 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, 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.

Kategorie tematyczne

• 47D09: Operator sine and cosine functions and higher-order Cauchy problems
• 47D03: Groups and semigroups of linear operators
• 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.
• 42C05: Orthogonal functions and polynomials, general theory
• 39B42: Matrix and operator equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 181
• Numer: 1
• Strony: 61-85

Daty

wydano

2007

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/sm181-1-5