Compact Perturbations Research Articles

Some new results on the property (R) under compact perturbations

The property (R) for a bounded linear operator T on a Hilbert space means that the points λ in the approximate point spectrum of T for which T − λI is upper semi-Browder are exactly the isolated points of the spectrum of T which are eigenvalues of finite multiplicity. For an operator T, if T + K satisfies the property (R) for all compact operators K, then T is said to have the stability of the property (R). In this paper, we give new criteria for the fulfillment of the stability of the property (R) for an operator. Moreover, the property (R) for complex symmetric operators is discussed.

Generalized weighted Hardy's inequalities with compact perturbations

Generalized weighted Hardy's inequalities with compact perturbations

Operators that are Weakly Coercive and a Compact Perturbation

ABSTRACT Let X, Y be two Banach spaces over K = ℝ \[\mathbb{K}=\mathbb{R}\] or ℂ \[\mathbb{C}\] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C 1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C 1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.

Endomorphisms of the Toeplitz algebra

This article describes all injective endomorphisms of the classical Toeplitz algebra. Their connection with endomorphisms of the algebra of continuous functions on the unit circle and with coverings over the unit circle was considered. It was shown that each non-unitary isometry V in the Toeplitz algebra determines the identity preserving endomorphism, as well as the class of its compact perturbations, i.e., identity non-preserving endomorphisms, defined by partial isometries {V P}, where P is a projection of finite codimension. The notions of T -equivalence of endomorphisms and T -equivalence up to a compact perturbation were introduced. An example was provided wherein the isometries are unitarily equivalent but the corresponding endomorphisms fall into different equivalence classes. Of all endomorphisms, the ones belonging to the class of Blaschke endomorphisms, which are analogous to endomorphisms of the discalgebra and generate unbranched coverings over the unit circle, were singled out.

Perturbations of gaps in the essential spectra of self-adjoint relations

This paper is concerned with the stability of gaps in the essential spectra of self-adjoint relations under non-negative relatively compact perturbation in Hilbert spaces. A stability result about semi-boundedness of self-adjoint relations under relatively bounded perturbation is given. It is shown that the gaps in the essential spectra of self-adjoint relations are invariant under non-negative relatively compact perturbation. The results obtained in the present paper generalize the corresponding results for linear operators to linear relations.

On growth and instability for semilinear evolution equations: an abstract approach

AbstractWe propose a new approach to the study of (nonlinear) growth and instability for semilinear abstract evolution equations with compact nonlinearities. We show, in particular, that compact nonlinear perturbations of linear evolution equations can be treated as linear ones as far as the growth of their solutions is concerned. We obtain exponential lower bounds of solutions for initial values from a dense set in resolvent or spectral terms. The abstract results are applied, in particular, to the study of energy growth for semilinear backward damped wave equations.

The parabolic algebra revisited

The parabolic algebra Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\cal A}_p}$$\\end{document} is the weakly closed operator algebra on L2(ℝ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L^2}(\\mathbb{R})$$\\end{document} generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions eiλx,λ≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${e^{i\\lambda x}},\\lambda\\ge 0$$\\end{document}. It is reflexive, with an invariant subspace lattice LatAp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\rm{Lat}}{{\\cal A}_p}$$\\end{document} which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). The structure of LatAp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\rm{Lat}}{{\\cal A}_p}$$\\end{document} is used to classify strongly irreducible isometric representations of the partial Weyl commutation relations. A formal generalisation of Arveson’s notion of a synthetic commutative subspace lattice is given for general subspace lattices, and it is shown that LatAp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\rm{Lat}}{{\\cal A}_p}$$\\end{document} is not synthetic relative to the H∞(ℝ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${H^\\infty}(\\mathbb{R})$$\\end{document} subalgebra of Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\cal A}_p}$$\\end{document}. Also, various new operator algebras, derived from isometric representations and from compact perturbations of Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\cal A}_p}$$\\end{document}, are defined and identified.

New Characterizations of the Jeribi Essential Spectrum

In this paper, we give several characterizations of the Jeribi essential spectrum of a bounded linear operator defined on a Banach space by using the notion of almost weakly compact operators. As a consequence, we prove the stability of the Jeribi essential spectrum under compact perturbations. Furthermore, some characterizations of the Jeribi essential spectra of 3×3 upper triangular block operator matrix are also given.

Coupling resonances and spectral properties of the product of resolvent and perturbation

Given a self-adjoint operator H0 and a relatively compact self-adjoint perturbation V, we study in some detail the spectral properties of the product (H0−z)−1V, z∈C. The eigenvalues of this product are related simply to those of the corresponding product in the situation when the operator H0 is perturbed to H0+sV for s∈C. For some numbers rz, the eigenvalues of (H0+sV−z)−1V, s∈C, are (s−rz)−1. We study the root spaces of the eigenvalues (s−rz)−1 and complex-analytic properties of the functions rz such as branching points. In particular, for a generic case, we give a variety of necessary and sufficient conditions for branching. The functions rz, called coupling resonances, are important in the spectral analysis of H0+rV for any real number r∈R. For instance, they afford a description of the spectral shift function (SSF) of the pair H0 and V, as well as the absolutely continuous and singular parts of the SSF. A thorough study of real-valued coupling resonances rλ for real λ outside of the essential spectrum was carried out in a recent work by the first author. Here we extend this study to the complex domain, motivated by the fact, which is well known in the case of a rank-one perturbation, that the behaviour of coupling resonances rz near the essential spectrum provides valuable information about the latter.

Uniform Linear Inviscid Damping and Enhanced Dissipation Near Monotonic Shear Flows in High Reynolds Number Regime (I): The Whole Space Case

We study the dynamics of the two dimensional Navier Stokes equations linearized around a strictly monotonic shear flow on $${\mathbb {T}}\times {\mathbb {R}}$$ . The main task is to understand the associated Rayleigh and Orr–Sommerfeld equations, under the natural assumption that the linearized operator around the monotonic shear flow in the inviscid case has no discrete eigenvalues. We obtain precise control of solutions to the Orr–Sommerfeld equations in the high Reynolds number limit, using the perspective that the nonlocal term can be viewed as a compact perturbation with respect to the main part that includes the small diffusion term. As a corollary, we give a detailed description of the linearized flow in Gevrey spaces (linear inviscid damping) that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The key difficulty is to accurately capture the behavior of the solution to Orr–Sommerfeld equations in the critical layer. In this paper we consider the case of shear flows on $${\mathbb {T}}\times {\mathbb {R}}$$ . The case of bounded channels poses significant additional difficulties, due to the presence of boundary layers, and will be addressed elsewhere.

On Nirenberg’s Problem for Compact Perturbations of Expansive Operator and an Application to p-Laplacian Type Equation with Nonmonotone Convection Function

On Nirenberg’s Problem for Compact Perturbations of Expansive Operator and an Application to p-Laplacian Type Equation with Nonmonotone Convection Function

Tele-Snap: a joint impedance estimation system using snap motor and openPose for remote rehabilitation diagnosis

We propose a remote joint impedance estimation system called Tele-snap for a rehabilitation diagnosis under the COVID-19 pandemic. Dynamic resistance of the human joint is essential physical information reflecting the motor function. The resistance is assessed based on the touching sensation of the doctor (physiotherapist), but the pandemic restricts such an in-person manner. Our proposing system aims to provide this physical information quantified by the joint impedance for a diagnosis in the telerehabilitation context. The proposed system employs a compact impulsive perturbation generator called the snap motor and a marker-less motion capture technology called the OpenPose. The subsystem installed in the patient's place is then simplified remarkably, which consists of the wearable snap motor and Raspberry Pi with a built-in camera module. The proposed system can collect the dataset for impedance estimation through the examiner's teleoperation of the snap motor and camera via a virtual private network, with no need for the operation by the patient. We verify the proposed system through an in-person experiment and then demonstrate the remote impedance estimation scheme.

A Data-Driven Approach to Viscous Fluid Mechanics: The Stationary Case

We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid’s viscosity in the mathematical model, we suggest directly using experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics (Kirchdoerfer and Ortiz in Comput Methods Appl Mech Eng 304:81–101, 2016; Conti et al. in Arch Ration Mech Anal 229:79–123, 2018). We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and {mathscr {A}}-quasiconvexity, we show a Gamma -convergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply not only to inertialess fluids but also to fluids with inertia. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.

Compact perturbations of Moore-Penrose invertible operators

Firstly, we study the stability of the Moore-Penrose invertibility of Hilbert space operators, establishing necessary and sufficient conditions for the invariance of the Moore-Penrose invertibility under small perturbations, compact perturbations and small compact perturbations. Secondly, we study the lifting problem of Moore-Penrose invertible elements of the Calkin algebra, showing that essentially Moore-Penrose invertible operators are small compact perturbations of Moore-Penrose invertible operators. Finally, we discuss the compactness of Moore-Penrose spectra, showing that each Hilbert space operator can be perturbed into an operator with nonempty compact Moore-Penrose spectrum by an arbitrarily small compact perturbation.

Quillen metric for singular families of Riemann surfaces with cusps and compact perturbation theorem

Quillen metric for singular families of Riemann surfaces with cusps and compact perturbation theorem

Growth-fragmentation equations with McKendrick–von Foerster boundary condition

The paper concerns the well-posedness and long-term asymptotics of growth–fragmentation equation with unbounded fragmentation rates and McKendrick–von Foerster boundary conditions. We provide three different methods of proving that there is a strongly continuous semigroup solution to the problem and show that it is a compact perturbation of the corresponding semigroup with a homogeneous boundary condition. This allows for transferring the results on the spectral gap available for the later semigroup to the one considered in the paper. We also provide sufficient and necessary conditions for the irreducibility of the semigroup needed to prove that it has asynchronous exponential growth. We conclude the paper by deriving an explicit solution to a special class of growth–fragmentation problems with McKendrick–von Foerster boundary conditions and by finding its Perron eigenpair that determines its long-term behaviour.

Complex symmetric operators with closed numerical ranges

A bounded linear operator T:H→H is said to be complex symmetric if there exists a conjugation C on H such that CT⁎C=T. In this paper, we study the numerical ranges of complex symmetric operators. We show that every complex symmetric operator T on H has a small compact perturbation being complex symmetric and having a closed numerical range. We also show that this holds for skew symmetric operators but fails to hold for unitary operators.

Solvability of discrete Helmholtz equations

Abstract We study the unique solvability of the discretized Helmholtz problem with Robin boundary conditions using a conforming Galerkin finite element method. Well-posedness of the discrete equations is typically investigated by applying a compact perturbation argument to the continuous Helmholtz problem so that a `sufficiently rich' discretization results in a `sufficiently small' perturbation of the continuous problem and well-posedness is inherited via Fredholm’s alternative. The qualitative notion `sufficiently rich', however, involves unknown constants and is only of asymptotic nature. Our paper is focussed on a fully discrete approach by mimicking the tools for proving well-posedness of the continuous problem directly on the discrete level. In this way, a computable criterion is derived, which certifies discrete well-posedness without relying on an asymptotic perturbation argument. By using this novel approach we obtain (a) new existence and uniqueness results for the $hp$-FEM for the Helmholtz problem, (b) examples for meshes such that the discretization becomes unstable (Galerkin matrix is singular) and (c) a simple checking Algorithm MOTZ `marching-of-the-zeros', which guarantees in an a posteriori way that a given mesh is certified for a well-posed Helmholtz discretization.

Cwikel estimates and negative eigenvalues of Schrödinger operators on noncommutative tori

In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension n ≥ 2. We use them to derive Cwikel–Lieb–Rozenblum inequalities and Lieb–Thirring inequalities for negative eigenvalues of fractional Schrödinger operators on noncommutative tori. The latter leads to a Sobolev inequality for noncommutative tori. On the way, we establish a “borderline version” of the abstract Birman–Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator with isolated 0-eigenvalue.

Spectral shift via “lateral” perturbation

We consider a compact perturbation H_0 = S + K_0^* K_0 of a self-adjoint operator S with an eigenvalue \lambda^\circ below its essential spectrum and the corresponding eigenfunction f . The perturbation is assumed to be “along” the eigenfunction f , namely K_0f=0 . The eigenvalue \lambda^\circ belongs to the spectra of both H_0 and S . Let S have \sigma more eigenvalues below \lambda^\circ than H_0 ; \sigma is known as the spectral shift at \lambda^\circ . We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue \lambda^\circ in the spectrum of H(K)=S + K^* K . We show that the eigenvalue as a function of K has a critical point at K=K_0 and the Morse index of this critical point is the spectral shift \sigma . A version of this theorem also holds for some non-positive perturbations.