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

Guy Métivier

Displaying similar documents to “Spectral Asymptotics for the $\bar{\partial}$ -Neumann Problem”

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

E. Porada (1971)

Colloquium Mathematicae

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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...

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 $(2)$ .

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...

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...

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.

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.

Schur Lemma and the Spectral Mapping Formula

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 $σ_{l} (S)$ = ${(λ (s))}_{s \in S} \in ℂ^{S} | {s - λ (s)}_{s \in S}$ generates a proper left ideal $.$ Using the Schur lemma and the Gelfand-Mazur theorem we prove that $σ_{l} (S)$ 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.

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 $F (ℂ, k)$ 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...

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...

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 $A u (x) = (\sum_{k = 1}^{N} e^{φ_{k}} T_{α_{k}}) u (x) = \sum_{k = 1}^{N} e^{φ_{k} (x)} u (α_{k} (x))$ , u ∈ C(X), where $φ_{k} \in C (X)$ and $α_{k} : X \to X$ 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 $l n (r (A)) = λ (φ) = m a x_{ν \in M e s} \sum_{k = 1}^{N} \int_{X} φ_{k} d ν_{k} - λ * (ν)$ , 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...

On the spectrum of the operator which is a composition of integration and substitution

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 $V_{ϕ} : f (x) \mapsto \int_{0}^{ϕ (x)} f (t) d t$ be defined on L₂[0,1]. We prove that $V_{ϕ}$ 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 $V_{ϕ}$ always equals 1.

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).

A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups

D. Müller, E. Prestini (2010)

Colloquium Mathematicae

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We define partial spectral integrals $S_{R}$ 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 $l o g (2 + L_{α}) f \in L ²$ , where $L_{α}$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R} f (x)$ converges a.e. to f(x) as R → ∞.

$Q$ -curvature and spectral invariants

Branson, Thomas

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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 $ℂ^{n + 1}$ 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.

Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory

Christophe Cuny (2010)

Studia Mathematica

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Let X be a closed subspace of $L^{p} (μ)$ , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $s u p_{n \in ℤ} | | U ⁿ | | < \infty$ . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $\sum_{n \geq 1} (U ⁿ f) / n^{1 - α}$ , 0 ≤ α < 1, in terms of $| | f + \dots + U^{n - 1} f {| |}_{p}$ , 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. ...

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...

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...

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...

Observability inequalities and measurable sets

Jone Apraiz, Luis Escauriaza, Gengsheng Wang, C. Zhang (2014)

Journal of the European Mathematical Society

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This paper presents two observability inequalities for the heat equation over $Ω \times (0, T)$ . In the first one, the observation is from a subset of positive measure in $Ω \times (0, T)$ , while in the second, the observation is from a subset of positive surface measure on $\partial Ω \times (0, T)$ . It also proves the Lebeau-Robbiano spectral inequality when $Ω$ is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

Localization of dominant eigenpairs and planted communities by means of Frobenius inner products

Dario Fasino, Francesco Tudisco (2016)

Czechoslovak Mathematical Journal

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We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$ . The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$ . Different choices for $X$ may be used, depending on the problem under investigation. In particular, we show that the choice where $X$ is the all-ones matrix allows to estimate the signature of the leading eigenvector of $A$ , generalizing previous results on Perron-Frobenius properties of matrices...

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...

Computation of some examples of Brown's spectral measure in free probability

Philippe Biane, Franz Lehner (2001)

Colloquium Mathematicae

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We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $u ₙ + u_{\infty}$ where uₙ and $u_{\infty}$ are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form $S_{α} + i S_{β}$ where $S_{α}$ and $S_{β}$ are free semicircular elements of variance α and β.

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 $ℱ_{♯ \cdot ♯} (X T X^{- 1})$ : X an invertible operator, where $ℱ_{♯ \cdot ♯} (T)$ is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if $_{♯ \cdot ♯} (T)$ is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then $d i s t (0, σ_{e} (T))$ = sup $_{♯ \cdot ♯} (X T X^{- 1})$ : X an invertible operator.

Displaying similar documents to “Spectral Asymptotics for the ∂ ¯ -Neumann Problem”

Displaying similar documents to “Spectral Asymptotics for the $\bar{\partial}$ -Neumann Problem”