On the spectral radius in
E. Porada (1971)
Colloquium Mathematicae
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E. Porada (1971)
Colloquium Mathematicae
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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 has the spectrum n + 2k = λ²: k a nonnegative integer. Let be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate 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...
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 , 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.
Dubi Kelmer, Peter Sarnak (2009)
Journal of the European Mathematical Society
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The existence of a strong spectral gap for quotients 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 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...
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 -submodule X̂ of ℬ(L²(G)) (where is the weak-* Haagerup tensor product ), define the concept of X̂-operator synthesis and prove that a...
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 vertices and clique number are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.
Victor S. Shulman, Yuriĭ V. Turovskii (2002)
Studia Mathematica
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The formula 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), is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.
Zhen Lin, Min Cai, Jiajia Wang (2024)
Czechoslovak Mathematical Journal
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Let be a simple connected undirected graph. The Laplacian spectral ratio of is defined as the quotient between the largest and second smallest Laplacian eigenvalues of , which is an important parameter in graph theory and networks. We obtain some bounds of the Laplacian spectral ratio in terms of the number of the spanning trees and the sum of powers of the Laplacian eigenvalues. In addition, we study the extremal Laplacian spectral ratio among trees with vertices, which improves...
Huicai Jia, Jing Lou (2024)
Czechoslovak Mathematical Journal
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For a set of graphs, an -factor of a graph is a spanning subgraph of , where each component of is contained in . It is very interesting to investigate the existence of factors in a graph with given minimum degree from the prospective of eigenvalues. We first propose a tight sufficient condition in terms of the -spectral radius for a graph involving minimum degree to contain a star factor. Moreover, we also present tight sufficient conditions based on the -spectral radius...
Antoni Wawrzyńczyk (2007)
Bulletin of the Polish Academy of Sciences. Mathematics
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Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula = generates a proper left idealUsing the Schur lemma and the Gelfand-Mazur theorem we prove that has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.
Dai Tamaki (2012)
Journal of the European Mathematical Society
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For a real central arrangement , Salvetti introduced a construction of a finite complex Sal which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement , the Salvetti complex Sal serves as a good combinatorial model for the homotopy type of the configuration space of points in , which is homotopy equivalent to the space of k little -cubes. Motivated by the importance of little cubes in homotopy theory, especially in...
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 bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is bounded? Both questions are related to a “uniform spectral radius” of the algebra, , introduced by Björk. Question (i) has an affirmative answer if and only if , and this result is extended to more general nonlinear extremal...
Krzysztof Zajkowski (2005)
Banach Center Publications
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We consider operators acting in the space C(X) (X is a compact topological space) of the form , u ∈ C(X), where and are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions . We prove that , where Mes is the set of all probability vectors of measures on X × 1,..., N and λ* is some convex lower-semicontinuous functional on...
Ignat Domanov (2008)
Studia Mathematica
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Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator be defined on L₂[0,1]. We prove that has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator always equals 1.
Francesco Belardo, Maurizio Brunetti, Adriana Ciampella (2021)
Czechoslovak Mathematical Journal
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A signed graph is a graph whose edges are labeled by signs. If has vertices, its spectral radius is the number , where are the eigenvalues of the signed adjacency matrix . Here we determine the signed graphs achieving the minimal or the maximal spectral radius in the classes and of unbalanced unicyclic graphs and unbalanced bicyclic graphs, respectively.
Michael Magee (2015)
Journal of the European Mathematical Society
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Let be a subgroup of an arithmetic lattice in . The quotient 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 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).
D. Müller, E. Prestini (2010)
Colloquium Mathematicae
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We define partial spectral integrals on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by , where is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that converges a.e. to f(x) as R → ∞.
Branson, Thomas
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Valentina Casarino, Paolo Ciatti (2009)
Studia Mathematica
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By using the notion of contraction of Lie groups, we transfer estimates for joint spectral projectors from the unit complex sphere in to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge’s work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.
Christophe Cuny (2010)
Studia Mathematica
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Let X be a closed subspace of , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like , 0 ≤ α < 1, in terms of , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...
Ajoy Jana, M. Thamban Nair (2019)
Czechoslovak Mathematical Journal
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It is known that the nonlinear nonhomogeneous backward Cauchy problem , with , where is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on and , that a solution of the above problem satisfies an integral equation involving the spectral representation of , which is also ill-posed. Spectral truncation...
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...
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 for all Borel subset Δ of the complex plane ℂ, where and 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...