Displaying similar documents to “A spectral gap theorem in SU ( d )

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 ( G ) -submodule X̂ of ℬ(L²(G)) (where V ( G ) is the weak-* Haagerup tensor product L ( G ) w * h L ( G ) ), define the concept of X̂-operator synthesis and prove that a...

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 j = 1 n [ ( x j + i y j ) ² + ( 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 ( d ) λ ϱ ( p ) | | u | | L ² ( 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...

Sufficient conditions on the existence of factors in graphs involving minimum degree

Huicai Jia, Jing Lou (2024)

Czechoslovak Mathematical Journal

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For a set { A , B , C , ... } of graphs, an { A , B , C , ... } -factor of a graph G is a spanning subgraph F of G , where each component of F is contained in { A , B , C , ... } . 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 Q -spectral radius for a graph involving minimum degree to contain a star factor. Moreover, we also present tight sufficient conditions based on the Q -spectral radius...

Spectral radius of operators associated with dynamical systems in the spaces C(X)

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 A u ( x ) = ( k = 1 N e φ k T α k ) u ( x ) = k = 1 N e φ k ( x ) u ( α k ( x ) ) , u ∈ C(X), where φ k C ( X ) and α k : X X 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 φ = ( φ k ) k = 1 N . We prove that l n ( r ( A ) ) = λ ( φ ) = m a x ν M e s k = 1 N X φ k d ν k - λ * ( ν ) , where Mes is the set of all probability vectors of measures ν = ( ν k ) k = 1 N on X × 1,..., N and λ* is some convex lower-semicontinuous functional on...

The Salvetti complex and the little cubes

Dai Tamaki (2012)

Journal of the European Mathematical Society

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For a real central arrangement 𝒜 , Salvetti introduced a construction of a finite complex Sal ( 𝒜 ) which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement 𝒜 k - 1 , the Salvetti complex Sal ( 𝒜 k - 1 ) serves as a good combinatorial model for the homotopy type of the configuration space F ( , k ) of k points in C , which is homotopy equivalent to the space C 2 ( k ) of k little 2 -cubes. Motivated by the importance of little cubes in homotopy theory, especially in...

Truncated spectral regularization for an ill-posed non-linear parabolic problem

Ajoy Jana, M. Thamban Nair (2019)

Czechoslovak Mathematical Journal

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It is known that the nonlinear nonhomogeneous backward Cauchy problem u t ( t ) + A u ( t ) = f ( t , u ( t ) ) , 0 t < τ with u ( τ ) = φ , where A 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 φ and f , that a solution of the above problem satisfies an integral equation involving the spectral representation of A , which is also ill-posed. Spectral truncation...

Spectral radius of weighted composition operators in L p -spaces

Krzysztof Zajkowski (2010)

Studia Mathematica

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We prove that for the spectral radius of a weighted composition operator a T α , acting in the space L p ( X , , μ ) , the following variational principle holds: l n r ( a T α ) = m a x ν M ¹ α , e X l n | a | d ν , where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, M ¹ α , e is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and -measurable function, where = n = 0 α - n ( ) . This considerably extends the range of validity of the above formula, which was previously known...

On the perturbation functions and similarity orbits

Haïkel Skhiri (2008)

Studia Mathematica

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We show that the essential spectral radius ϱ e ( T ) of T ∈ B(H) can be calculated by the formula ϱ e ( T ) = inf · ( X T X - 1 ) : X an invertible operator, where · ( T ) is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if · ( T ) is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then d i s t ( 0 , σ e ( T ) ) = sup · ( X T X - 1 ) : X an invertible operator.

Matrix coefficients, counting and primes for orbits of geometrically finite groups

Amir Mohammadi, Hee Oh (2015)

Journal of the European Mathematical Society

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Let G : = SO ( n , 1 ) and Γ ( n - 1 ) / 2 for n = 2 , 3 and when δ > n - 2 for n 4 , we obtain an effective archimedean counting result for a discrete orbit of Γ in a homogeneous space H G where H is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family { T H G } of compact subsets, there exists η > 0 such that # [ e ] Γ T = ( T ) + O ( ( T ) 1 - η ) for an explicit measure on H G which depends on Γ . We also apply the affine sieve and describe the distribution of almost primes on orbits of Γ in arithmetic...

Truncation and Duality Results for Hopf Image Algebras

Teodor Banica (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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Associated to an Hadamard matrix H M N ( ) is the spectral measure μ ∈ [0,N] of the corresponding Hopf image algebra, A = C(G) with G S N . We study a certain family of discrete measures μ r [ 0 , N ] , coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type 0 N ( x / N ) p d μ r ( x ) = 0 N ( x / N ) r d ν p ( x ) , where μ r , ν r are the truncations of the spectral measures μ,ν associated to H , H t . We also prove, using these truncations μ r , ν r , that for any deformed Fourier matrix H = F M Q F N we have μ = ν.

Spectral radius and Hamiltonicity of graphs with large minimum degree

Vladimir Nikiforov (2016)

Czechoslovak Mathematical Journal

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Let G be a graph of order n and λ ( G ) the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in G . One of the main results of the paper is the following theorem: Let k 2 , n k 3 + k + 4 , and let G be a graph of order n , with minimum degree δ ( G ) k . If λ ( G ) n - k - 1 , then G has a Hamiltonian cycle, unless G = K 1 ( K n - k - 1 + K k ) or G = K k ( K n - 2 k + K ¯ k ) .

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 ) f x ( R F ) d μ ( x ) = h ( R F ) . (1) They involve functions of the kind X x f x ( R F ) B ( F ) , where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

A spectral bound for graph irregularity

Felix Goldberg (2015)

Czechoslovak Mathematical Journal

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The imbalance of an edge e = { u , v } in a graph is defined as i ( e ) = | d ( u ) - d ( v ) | , where d ( · ) is the vertex degree. The irregularity I ( G ) of G is then defined as the sum of imbalances over all edges of G . This concept was introduced by Albertson who proved that I ( G ) 4 n 3 / 27 (where n = | V ( G ) | ) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves...

Transferring L p eigenfunction bounds from S 2 n + 1 to hⁿ

Valentina Casarino, Paolo Ciatti (2009)

Studia Mathematica

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By using the notion of contraction of Lie groups, we transfer L p - L ² estimates for joint spectral projectors from the unit complex sphere S 2 n + 1 in n + 1 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.

On the bounds of Laplacian eigenvalues of k -connected graphs

Xiaodan Chen, Yaoping Hou (2015)

Czechoslovak Mathematical Journal

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Let μ n - 1 ( G ) be the algebraic connectivity, and let μ 1 ( G ) be the Laplacian spectral radius of a k -connected graph G with n vertices and m edges. In this paper, we prove that μ n - 1 ( G ) 2 n k 2 ( n ( n - 1 ) - 2 m ) ( n + k - 2 ) + 2 k 2 , with equality if and only if G is the complete graph K n or K n - e . Moreover, if G is non-regular, then μ 1 ( G ) < 2 Δ - 2 ( n Δ - 2 m ) k 2 2 ( n Δ - 2 m ) ( n 2 - 2 n + 2 k ) + n k 2 , where Δ stands for the maximum degree of G . Remark that in some cases, these two inequalities improve some previously known results.

On open maps and related functions over the Salbany compactification

Mbekezeli Nxumalo (2024)

Archivum Mathematicum

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Given a topological space X , let 𝒰 X and η X : X 𝒰 X denote, respectively, the Salbany compactification of X and the compactification map called the Salbany map of X . For every continuous function f : X Y , there is a continuous function 𝒰 f : 𝒰 X 𝒰 Y , called the Salbany lift of f , satisfying ( 𝒰 f ) η X = η Y f . If a continuous function f : X Y has a stably compact codomain Y , then there is a Salbany extension F : 𝒰 X Y of f , not necessarily unique, such that F η X = f . In this paper, we give a condition on a space such that its Salbany map is open. In...