Eigenvalues and Entropy Moduli of Operators in Interpolation Spaces

The paper approaches in an abstract way the spectral theory of operators in abstract interpolation spaces. We introduce entropy numbers and spectral moduli of operators, and prove a relationship between them and eigenvalues of operators. We also investigate interpolation variants of the moduli, and offer a contribution to the theory of eigenvalues of operators. Specifically, we prove an interpolation version of the celebrated Carl–Triebel eigenvalue inequality. Based on these results we are able to prove interpolation estimates for single eigenvalues as well as for geometric means of absolute values of the first n eigenvalues of operators. In particular, some of these estimates may be regarded as generalizations of the classical spectral radius formula. We give applications of our results to the study of interpolation estimates of entropy numbers as well as of the essential spectral radius of operators in interpolation spaces.     

一类具抽象边界条件的迁移算子的谱分布

利用线性算子半群理论,研究了板几何中具抽象边界条件的各向异性、连续能量、非均匀介质的迁移方程．在假设边界算子日部分光滑和扰动算子K正则的条件下,采用豫解方法,得到了该迁移算子A的谱在区域Г中由至多可数个具有限代数重数的离散本征值组成等结果．     

Universal inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.     

Weyl numbers and eigenvalues of abstract summing operators

We estimate Weyl numbers and eigenvalues of operators via studying their abstract summing norms. In particular we prove estimates of these summing norms for abstract interpolation Lorentz spaces. For this we combine factorization theorems with estimates of concavity constants. Finally we apply our general eigenvalue results to integral operators with kernels of weakly singular type. We obtain asymptotically optimal estimates which extend the well-known classical results.     

Eigenvalues for Radially Symmetric Fully Nonlinear Operators

In this paper we present an elementary theory about the existence of eigenvalues for fully nonlinear radially symmetric 1-homogeneous operators. A general theory for first eigenvalues and eigenfunctions of 1-homogeneous fully nonlinear operators exists in the framework of viscosity solutions. Here we want to show that for the radially symmetric operators or in the one dimensional case a much simpler theory, based on ode and degree theory arguments, can be established. We obtain the complete set of eigenvalues and eigenfunctions characterized by the number of zeroes.     

A bound for ratios of eigenvalues of Schrödinger operators with single-well potentials

For Schrödinger operators with nonnegative single-well potentials ratios of eigenvalues are extremal only in the case of zero potential. To prove this, we investigate some monotonicity properties of Prüfer-type variables.

     

带势加权散度形式的 Grushin 型退化椭圆算子的 Dirichlet 特征值的上界*

本文考虑了带有位势的散度形式的 Grushin 型退化椭圆算子的 Dirichlet 加权特征值的估计.利 用傅里叶变换的方法得到了特征值的精确下界估计.然后通过试验函数的方法得到了特征值上界的杨型 不等式.     

Twistor Operators and Eigenvalues of the Dirac Operator on Compact Quaternion-Kähler Spin Manifolds

Quaternion-Kähler twistor operators are introduced. Using these operators with the Lichnerowicz formula, we get lower bounds for the square of the eigenvalues of the Dirac operator in terms of the eigenvalues of the fundamental 4-form.     

On the Asymptotics of Eigenvalues of an Analytical Pencil of Operators

We study the asymptotic behavior of eigenvalues of a perturbed pencil of operators in Banach spaces. We prove theorems on branching of an isolated eigenvalue into a finite number of simple eigenvalues. In addition, we deduce conditions for obtaining the asymptotics of a given form.     

带权散度型椭圆算子的低阶特征值估计

本文首先给出紧致带边(边界可以为空集)光滑度量测度空间上带权散度型算子的低阶特征值的一个一般不等式,通过使用这个一般不等式,可以得到光滑度量测度空间中有界连通区域上带权散度型算子的低阶特征值的一些万有不等式.     

Isoparametric finite-element approximation of a Steklov eigenvalue problem

We study the isoparametric variant of the finite-element method(FEM) for an approximation of Steklov eigenvalue problems forsecond-order, selfadjoint, elliptic differential operators.Error estimates for eigenfunctions and eigenvalues are derived.We prove the same estimate for eigenvalues as that obtainedin the case of conforming finite elements provided that theboundary of the domain is well approximated. Some algorithmicaspects arising from the FE isoparametric discretization ofthe Steklov problems are analysed. We finish this paper withnumerical results confirming the considered theory.     

Some inequalities involving eigenvalues of the Neumann Laplacian

This paper is concerned with the eigenvalues of the Neumann Laplacian on various classes of domains of given measure: simply‐connected Lipschitz planar domains, n‐sided planar polygons and smooth N‐dimensional domains. In each case, we consider some quantities involving low eigenvalues of the Neumann Laplacian for which we obtain new inequalities. Moreover, we sharpen a universal bound derived by M. Ashbaugh and R. Benguria for sum of reciprocal of Neumann eigenvalues. Our investigations make use of some properties of conformal mappings, Bessel functions, symmetric domains or some isoperimetric inequalities for moments of inertia. Copyright © 2013 John Wiley & Sons, Ltd.     

The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control

In this paper, we give an abstract condition of Riesz basisgeneration for discrete operators in Hilbert spaces, from whichwe show that the generalized eigenfunctions of a Euler–Bernoullibeam equation with boundary linear feedback control form a Rieszbasis for the state Hilbert space. As an consequence, the asymptoticexpression of eigenvalues together with exponential stabilityare readily presented.     

Lp resolvent estimates for magnetic Schrödinger operators with unbounded background fields

We prove L^p and smoothing estimates for the resolvent of magnetic Schrödinger operators. We allow electromagnetic potentials that are small perturbations of a smooth, but possibly unbounded background potential. As an application, we prove an estimate on the location of eigenvalues of magnetic Schrödinger and Pauli operators with complex electromagnetic potentials.     

Inequalities for Eigenvalues of a System of Equations of Elliptic Operator in Weighted Divergence Form on Metric Measure Space

Let A be a symmetric and positive definite(1, 1) tensor on a bounded domain Ω in an ndimensional metric measure space■. In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form■,where ■, α is a nonnegative constant and u is a vector-valued function.Some universal inequalities for eigenvalues of this problem are established. Moreover, as applications of these results, we give some estimates for the upper bound of ?k+1 and the gap of ?k+1-?k in terms of the first k eigenvalues. Our results contain some results for the Lam′e system and a system of equations of the drifting Laplacian.     

Some results,concerning the variation of eigenvalues,when these depend on a real parameter

In this article we deal with a Hamiltonial of the form H(v) = Ho + A(v) where Ho is a self-adjoint bounded or unbounded operator on a Hilbert space and A(v) is a bounded self-adjoint perturbation depending on a real parameter v. In quantum mechanics a variety of results has been obtained by taking formally the derivative of the eigenvectors and eigenvalues of H(v).The differentiability of the eigenvectors and eigenvalues has been rigorously proved under several assumptions. Among these assumptions is the assumption that the eigenvalues are simple and the assumption that the perturbation A(v) is a uniformly bounded self-adjoint operator. A part of this article is dealing with examples, which show that these two assumptions are essential. The rest of this article is devoted to different applications concerning asymptotic relations of eigenvalues and a result for the solutions of the equation d_y/d_t= M(t)y in an abstract infinite dimensional Hilbert space, where iM(t)(1²=-1) is self-adjoint for every t in an interval. This result finds a succesful application to the theory of Toda and Langmuir lattices.     

On One Class of Operators Associated with Differential Equations of Fractional Order

We carry out spectral analysis of one class of integral operators associated with fractional order differential equations applicable in mechanics. We establish connection between the eigenvalues of these operators and the zeros of Mittag-Leffler type functions. We give sufficient conditions for complete nonselfadjointness.     

On Riesz Means of Eigenvalues

In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the Riemann–Liouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Dirichlet Laplacians as well as certain Schrödinger operators. At the heart of these inequalities are calculations of commutators of operators, sum rules, and monotonic properties of Riesz means. In the course of developing these inequalities we prove new bounds for the partition function and the spectral zeta function (cf. Corollaries 3.5–3.7) and conjecture about additional bounds.     

Estimates for the Eigenvalues of Matrices

For matrices whose eigenvalues are real (such as Hermitian or real symmetric matrices), we derive simple explicit estimates for the maximal (λ_max) and the minimal (λ_min) eigenvalues in terms of determinants of order less than 3. For 3 × 3 matrices, we derive sharper estimates, which use det A but do not require to solve cubic equations.     

带权Paneitz算子和带权圆盘振动问题的特征值不等式

本文研究光滑度量测度空间上带权Paneitz算子的闭特征值问题和带权圆盘振动问题,给出Euclid空间、单位球面、射影空间和一般Riemann流形的n维紧子流形的权重Paneitz箅子和带权圆盘振动问题的前n个特征值上界估计.进一步地,本文给出带权Ricci曲率有界的紧致度量测度空间上带权圆盘振动问题的第一特征值的下界...