We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm grows at least quadratically with the distance from this point. This provides a new proof not using the maximum principle that the resolvent norm of the considered class cannot have local maxima. Finally, we give new criteria for the existence of local non-degenerate minima of the resolvent norm and provide examples of (un)bounded non-normal operators having this property.
In this paper, we show that the two concepts of generalised norm resolvent convergence introduced by Weidmann and the first author of this paper are equivalent. We also focus on the convergence speed and provide conditions under which the convergence speed is the same for both concepts. We illustrate the abstract results by a large number of examples.
This paper is devoted to dimensional reductions via the norm resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different PDE problems from various areas of mathematical physics; all are analysed in a unified manner now, known results are recovered and new ones established.
We prove norm estimates for the difference of resolvents of operators and their discrete counterparts, embedded into the continuum using biorthogonal Riesz sequences. The estimates are given in the operator norm for operators on square integrable functions, and depend explicitly on the mesh size for the discrete operators. The operators are a sum of a Fourier multiplier and a multiplicative potential. The Fourier multipliers include the fractional Laplacian and the pseudo-relativistic free Hamiltonian. The potentials are real, bounded, and Hölder continuous. As a side-product, the Hausdorff di…
We show a norm convergence result for the Laplacian on a class of post-critically finite fractals with arbitrary Borel regular probability measure which can be approximated by a sequence of finite-dimensional graph Laplacians with corresponding discrete probability measures. As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm.
Eigenvalue estimates for the resolvent of a non-normal matrix
研究了非正规矩阵的谱与其预解式范数之间的关系。我们提供谱估计的矩阵的最大奇异值是有界的$1$（所谓的希尔伯特空间压缩）和幂有界矩阵的预解式。在第一种情况下，我们的估计是最佳的，我们提出了明确的矩阵，实现平等的界限。这一结果恢复和推广了E. B. Davies和B. Simon在单位圆上正交多项式的研究中。在幂有界矩阵的情况下，我们实现了迄今为止最强的估计。我们的结果统一了以前的方法，其中的预解估计在某些限制区域的复平面。为了实现我们的估计，我们涉及到的问题，界定一个矩阵的函数的范数的Nevanlinna-Pick插值问题在相应的函数空间。在希尔伯特空间收缩的情况下，这个问题是连接到理论的压缩移位算子，我们有助于提供明确的矩阵表示，这样的运营商。最后，我们应用我们的结果来研究一个经典的或量子马尔可夫链的平稳状态相对于扰动的转移阵的敏感性。
Since the publication of the important work of Rauch and Taylor (Potential and scattering theory on wildly perturbed domains, JFA, 1975) a lot has been done to analyse wild perturbations of the Laplace operator. Here we present results concerning the norm convergence of the resolvent. We consider a (not necessarily compact) manifold with many small balls removed, the number of balls can increase as the radius is shrinking, the number of balls can also be infinite. If the distance of the balls shrinks less fast than the radius, then we show that the Neumann Laplacian converges to the unperturbe…
We consider a general second order matrix operator in a multi-dimensional domain subject to a classical boundary condition. This operator is perturbed by a first order differential operator, the coefficients of which depend arbitrarily on a small multi-dimensional parameter. We study the existence of a limiting (homogenized) operator in the sense of the norm resolvent convergence for such perturbed operator. The first part of our main results states that the norm resolvent convergence is equivalent to the convergence of the coefficients in the perturbing operator in certain space of multiplier…
Resolvent estimates for one-dimensional Dirac operators with imaginary potentials
Institute of Applied Mathematics, Graz University of Technology（格拉茨工业大学应用数学研究所）；Analytical and Algebraic Methods in Optimization Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University（同德大学数学与统计学院分析与代数优化方法研究组）
In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2π\mathbb{Z})^2$ with Hölder dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, …
We present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature, the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the oper…
