On the spectral multiplicity of a direct sum of operators

M. T. Karaev

Displaying similar documents to “On the spectral multiplicity of a direct sum of operators”

Formulae for joint spectral radii of sets of operators

Victor S. Shulman, Yuriĭ V. Turovskii (2002)

Studia Mathematica

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The formula $ϱ (M) = m a x ϱ_{χ} (M), r (M)$ is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), $ϱ_{χ} (M)$ is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.

On the joint spectral radius

Vladimír Müller (1997)

Annales Polonici Mathematici

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We prove the $_{p}$ -spectral radius formula for n-tuples of commuting Banach algebra elements

Spectral mapping inclusions for the Phillips functional calculus in Banach spaces and algebras

Eva Fašangová, Pedro J. Miana (2005)

Studia Mathematica

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We investigate the weak spectral mapping property (WSMP) $\bar{μ ̂ (σ (A))} = σ (μ ̂ (A))$ , where A is the generator of a ₀-semigroup in a Banach space X, μ is a measure, and μ̂(A) is defined by the Phillips functional calculus. We consider the special case when X is a Banach algebra and the operators $e^{A t}$ , t ≥ 0, are multipliers.

Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs

Kinkar Ch. Das, Muhuo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with $n$ vertices and clique number $ω$ $(2 \leq ω \leq n)$ are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.

The spectral decomposition of weighted shifts and the $A^{p}$ condition

Earl Berkson, T. A. Gillespie (1990)

Colloquium Mathematicae

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Spectral Asymptotics for the $\bar{\partial}$ -Neumann Problem

Guy Métivier (1981-1982)

Publications mathématiques et informatique de Rennes

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On the zero set of the Kobayashi-Royden pseudometric of the spectral unit ball

Nikolai Nikolov, Pascal J. Thomas (2008)

Annales Polonici Mathematici

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Given A∈ Ωₙ, the n²-dimensional spectral unit ball, we show that if B is an n×n complex matrix, then B is a “generalized” tangent vector at A to an entire curve in Ωₙ if and only if B is in the tangent cone $C_{A}$ to the isospectral variety at A. In the case of Ω₃, the zero set of the Kobayashi-Royden pseudometric is completely described.

Bounded point evaluations for multicyclic operators

M. EL Guendafi, M. Mbekhta, E. H. Zerouali (2005)

Banach Center Publications

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Let T be a multicyclic operator defined on some Banach space. Bounded point evaluations and analytic bounded point evaluations for T are defined to generalize the cyclic case. We extend some known results on cyclic operators to the more general setting of multicyclic operators on Banach spaces. In particular we show that if T satisfies Bishop’s property (β), then $ℬ_{a} = ℬ ∖ σ_{a p} (T)$ . We introduce the concept of analytic structures and we link it to different spectral quantities. We apply this concept...

On the spectral radius in $L_{1} (G)$

E. Porada (1971)

Colloquium Mathematicae

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Strong spectral gaps for compact quotients of products of $PSL (2, ℝ)$ )

Dubi Kelmer, Peter Sarnak (2009)

Journal of the European Mathematical Society

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The existence of a strong spectral gap for quotients $Γ ∖ G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan–Selberg conjectures. If $G$ has no compact factors then for general lattices a spectral gap can still be established, but there is no uniformity and no effective bounds are known. This note is concerned with the spectral...

A spectral gap property for subgroups of finite covolume in Lie groups

Bachir Bekka, Yves Cornulier (2010)

Colloquium Mathematicae

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Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G. We show that the unitary representation $λ_{G / H}$ of G on L²(G/H) has a spectral gap, that is, the restriction of $λ_{G / H}$ to the orthogonal complement of the constants in L²(G/H) does not have almost invariant vectors. This answers a question of G. Margulis. We give an application to the spectral geometry of locally symmetric Riemannian spaces of infinite volume.

A spectral gap theorem in SU $(d)$

Jean Bourgain, Alex Gamburd (2012)

Journal of the European Mathematical Society

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We establish the spectral gap property for dense subgroups of SU $(d)$ $(d \geq 2)$ , generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU $(2)$ .

Spectral projections for the twisted Laplacian

Herbert Koch, Fulvio Ricci (2007)

Studia Mathematica

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Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = - 1 / 2 \sum_{j = 1}^{n} [(\partial_{x_{j}} + i y_{j}) ² + (\partial_{y_{j}} - i x_{j}) ²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $| | P_{λ} {u | |}_{L^{p} (ℝ^{d})} ≲ λ^{ϱ (p)} {| | u | |}_{L ² (ℝ^{d})}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p...

Spectral synthesis and operator synthesis

K. Parthasarathy, R. Prakash (2006)

Studia Mathematica

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Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty} (G)$ -submodule X̂ of ℬ(L²(G)) (where $V^{\infty} (G)$ is the weak-* Haagerup tensor product $L^{\infty} (G) \otimes_{w * h} L^{\infty} (G)$ ), define the concept of X̂-operator synthesis and prove that a...

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

On the weak decomposition property $(δ_{w})$

El Hassan Zerouali, Hassane Zguitti (2005)

Studia Mathematica

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We study a new class of bounded linear operators which strictly contains the class of bounded linear operators with the decomposition property (δ) or the weak spectral decomposition property (weak-SDP). We treat general local spectral properties for operators in this class and compare them with the case of operators with (δ).

Uniform spectral radius and compact Gelfand transform

Alexandru Aleman, Anders Dahlner (2006)

Studia Mathematica

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We consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x ↦ x̂ are: (i) Is $K_{ν} = s u p {| | (e - x)}^{- 1} {| |}_{p} {: x \in A, | | x | |}_{p} \leq 1, m a x | x ̂ | \leq ν$ bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is $C_{δ} = s u p | | x^{- 1} {| |}_{p} {: x \in A, | | x | |}_{p} \leq 1, m i n | x ̂ | \geq δ$ bounded? Both questions are related to a “uniform spectral radius” of the algebra, $r_{\infty} (A)$ , introduced by Björk. Question (i) has an affirmative answer if and only if $r_{\infty} (A) < 1$ , and this result is extended to more general nonlinear extremal...

$^{\infty}$ -vectors and boundedness

Jan Stochel, F. H. Szafraniec (1997)

Annales Polonici Mathematici

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The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its $^{\infty}$ -vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

Linear maps on Mₙ(ℂ) preserving the local spectral radius

Abdellatif Bourhim, Vivien G. Miller (2008)

Studia Mathematica

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Let x₀ be a nonzero vector in ℂⁿ. We show that a linear map Φ: Mₙ(ℂ) → Mₙ(ℂ) preserves the local spectral radius at x₀ if and only if there is α ∈ ℂ of modulus one and an invertible matrix A ∈ Mₙ(ℂ) such that Ax₀ = x₀ and $Φ (T) = α A T A^{- 1}$ for all T ∈ Mₙ(ℂ).

Absolutely continuous and singular spectral shift functions

Nurulla Azamov

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Given a self-adjoint operator H₀, a self-adjoint trace-class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and cokernel, using the limiting absorption principle an explicit set Λ(H₀;F) ⊂ ℝ of full Lebesgue measure is defined, such that for all λ ∈ Λ(H₀+rV;F) ∩ Λ(H₀;F), where r ∈ ℝ, the wave $w_{\pm} (λ; H ₀ + r V, H ₀)$ and the scattering matrices S(λ;H₀+rV,H₀) can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including...

Subnormality and cyclicity

Franciszek Hugon Szafraniec (2005)

Banach Center Publications

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For an unbounded operator S the question whether its subnormality can be built up from that of every $S_{f}$ , the restriction of S to a cyclic space generated by f in the domain of S, is analyzed. Though the question at large has been left open some partial results are presented and a possible way to prove it is suggested as well.

Some results on $A_{p} (G)$ submodules of $P M_{p} (G)$

Edmond E. Granirer (1987)

Colloquium Mathematicae

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A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F (x) : = F ̃ x^{(2 n + 1)}$ where $F ̃ : X^{2 n + 1} \to Y$ is a continuous (2n+1)-linear operator.

On a binary relation between normal operators

Takateru Okayasu, Jan Stochel, Yasunori Ueda (2011)

Studia Mathematica

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The main goal of this paper is to clarify the antisymmetric nature of a binary relation ≪ which is defined for normal operators A and B by: A ≪ B if there exists an operator T such that $E_{A} (Δ) \leq T * E_{B} (Δ) T$ for all Borel subset Δ of the complex plane ℂ, where $E_{A}$ and $E_{B}$ are spectral measures of A and B, respectively (the operators A and B are allowed to act in different complex Hilbert spaces). It is proved that if A ≪ B and B ≪ A, then A and B are unitarily equivalent, which shows that the relation ≪ is...

Geometry of the spectral semidistance in Banach algebras

Gareth Braatvedt, Rudi Brits (2014)

Czechoslovak Mathematical Journal

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Let $A$ be a unital Banach algebra over $ℂ$ , and suppose that the nonzero spectral values of $a$ and $b \in A$ are discrete sets which cluster at $0 \in ℂ$ , if anywhere. We develop a plane geometric formula for the spectral semidistance of $a$ and $b$ which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that $a$ and $b$ are quasinilpotent equivalent...

Quantitative spectral gap for thin groups of hyperbolic isometries

Michael Magee (2015)

Journal of the European Mathematical Society

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Let $Λ$ be a subgroup of an arithmetic lattice in $SO (n + 1, 1)$ . The quotient $ℍ^{n + 1} / Λ$ has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense $Λ$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

(Non-)amenability of ℬ(E)

Volker Runde (2010)

Banach Center Publications

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In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ (E) = (E) + ℂ i d_{E}$ . Still, ℬ(ℓ²) is...

Integral equalities for functions of unbounded spectral operators in Banach spaces

Benedetto Silvestri

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The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form $g (R_{F}) \int f_{x} (R_{F}) d μ (x) = h (R_{F})$ . (1) They involve functions of the kind $X ∋ x \mapsto f_{x} (R_{F}) \in B (F)$ , where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

Estimation of the Szlenk index of Banach spaces via Schreier spaces

Ryan Causey (2013)

Studia Mathematica

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For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index $ω^{α + 1}$ which is universal for the class of separable Banach spaces with Szlenk index not exceeding $ω^{α}$ . Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.

Growth and smooth spectral synthesis in the Fourier algebras of Lie groups

Jean Ludwig, Lyudmila Turowska (2006)

Studia Mathematica

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Let G be a Lie group and A(G) the Fourier algebra of G. We describe sufficient conditions for complex-valued functions to operate on elements u ∈ A(G) of certain differentiability classes in terms of the dimension of the group G. Furthermore, generalizing a result of Kirsch and Müller [Ark. Mat. 18 (1980), 145-155] we prove that closed subsets E of a smooth m-dimensional submanifold of a Lie group G having a certain cone property are sets of smooth spectral synthesis. For such sets we...