A spectral theory for solvable Lie algebras of operators

Boasso, Enrico; Larotonda, Angel

doi:10.2140/pjm.1993.158.15

Cited by 18 publications

(70 citation statements)

References 6 publications

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“…Now we state the definition of the Taylor and the S lodkowski joint spectra; see [5], [7], [16], [20] and [21]. We follow the notation of [21, Definition 2.11.1].…”

Section: The Taylor the S Lodkowski And The Split Joint Spectramentioning

confidence: 99%

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Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras

Boasso

2003

Dissertationes Math.

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Given two complex Banach spaces X 1 and X 2 , a tensor product X 1⊗ X 2 of X 1 and X 2 in the sense of [14], two complex solvable finite dimensional Lie algebras L 1 and L 2 , and two representations ρ i : L i → L(X i ) of the algebras, i = 1, 2, we consider the Lie algebra L = L 1 × L 2 , and the tensor product representation of L, ρ : L → L(X 1⊗ X 2 ), ρ = ρ 1 ⊗ I + I ⊗ ρ 2 . In this work we study the S lodkowski and the split joint spectra of the representation ρ, and we describe them in terms of the corresponding joint spectra of ρ 1 and ρ 2 . Moreover, we study the essential S lodkowski and the essential split joint spectra of the representation ρ, and we describe them by means of the corresponding joint spectra and the corresponding essential joint spectra of ρ 1 and ρ 2 . In addition, with similar arguments we describe all the above-mentioned joint spectra for the multiplication representation in an operator ideal between Banach spaces in the sense of [14]. Finally, we consider nilpotent systems of operators, in particular commutative, and we apply our descriptions to them.

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“…Now we state the definition of the Taylor and the S lodkowski joint spectra; see [5], [7], [16], [20] and [21]. We follow the notation of [21, Definition 2.11.1].…”

Section: The Taylor the S Lodkowski And The Split Joint Spectramentioning

confidence: 99%

“…Indeed, sincef ∈ σ δ,k,e (ρ | I) ⊆ σ δ,k (ρ | I),f is a character of I, i.e.,f (I 2 ) = 0. However, since I is an ideal of L, by the projection property of the joint spectrum σ δ,k (ρ) (see [5,Theorem 4.5], [20,Theorem 3.4] and [21, Satz 2.11.5]), there is f ∈ σ δ,k,e (ρ) such that f | I =f .…”

Section: The Fredholm Joint Spectramentioning

confidence: 99%

Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras

Boasso

2003

Dissertationes Math.

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“…[8]- [12]) спектральной теории нильпотентных операторных алгебр Ли открывают возможности применения метода Тейлора к универсальной обертывающей алгебре B = U (g) конечномерной нильпотент-ной алгебры Ли g. В частности, можно дать определение спектра Тейлора σ(g, X) банахова U (g)-модуля X, обладающего свойством спектрального отоб-ражения по отношению к некоммутативным полиномам.…”

Section: § 1 введениеunclassified

Спектр Тейлора И Трансверсальность Для Операторной Алгебры Гейзенберга

Dosi¹,

Dosi²

2010

Матем. сб.

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Спектр Тейлора и трансверсальность для операторной алгебры ГейзенбергаРассмотрены проблемы некоммутативного голоморфного функцио-нального исчисления для банаховых модулей над конечномерными ниль-потентными алгебрами Ли. Основной результат устанавливает свойство трансверсальности алгебр некоммутативных голоморфных функций по отношению к спектру Тейлора семейства ограниченных линейных опера-торов, порождающего алгебру Гейзенберга.Библиография: 25 названий.Ключевые слова: голоморфная функция от элементов алгебры Ли, спектр Тейлора, свойство трансверсальности, обращающее пополнение Фреше.

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“…and λ ∈ σ(id G ) (see [5], [6] and Theorem 3.6.7 of [15]). Here, of course, we denote by id G the natural inclusion map of the vector space G in B(X ), which can be viewed as a representation of G. …”

Section: Where (λ ⊗ λ)(T S) = λ(T ) + λ(S) For T S ∈ G Opmentioning

confidence: 99%

Analytic joint spectral radius in a solvable Lie algebra of operators

Beltiţă

2001

Studia Math.

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Abstract.We introduce the concept of analytic spectral radius for a family of operators indexed by some finite measure space. This spectral radius is compared with the algebraic and geometric spectral radii when the operators belong to some finite-dimensional solvable Lie algebra. We describe several situations when the three spectral radii coincide. These results extend well known facts concerning commuting n-tuples of operators.

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A spectral theory for solvable Lie algebras of operators

Abstract: The main objective of this paper is to develop a notion of joint spectrum for complex solvable Lie algebras of operators acting on a Banach space, which generalizes Taylor joint spectrum (T.J.S.) for several commuting operators.

Cited by 18 publications

References 6 publications

Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras

Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras

Спектр Тейлора И Трансверсальность Для Операторной Алгебры Гейзенберга

Analytic joint spectral radius in a solvable Lie algebra of operators

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