Compact Perturbations Research Articles

On algebras generated by Toeplitz operators and their representations

We study Banach and C⁎-algebras generated by Toeplitz operators acting on weighted Bergman spaces Aλ2(B2) over the complex unit ball B2⊂C2. Our key point is an orthogonal decomposition of Aλ2(B2) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space Aμ2(D) over the complex unit disk D. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on Aμ2(D).The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L(Aμ2(D)) are of particular interest. In this paper we discuss various examples. In the case of S=C(D‾) and S=C(D‾)⊗L∞(0,1) we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.

ON n-*-PARANORMAL OPERATORS

A Hilbert space operator <TEX>$T{\in}{\mathfrak{B}}(\mathfrak{H})$</TEX> is said to be n-*-paranormal, <TEX>$T{\in}C(n)$</TEX> for short, if <TEX>${\parallel}T^*x{\parallel}^n{\leq}{\parallel}T^nx{\parallel}\;{\parallel}x{\parallel}^{n-1}$</TEX> for all <TEX>$x{\in}{\mathfrak{H}}$</TEX>. We proved some properties of class C(n) and we proved an asymmetric Putnam-Fuglede theorem for n-*-paranormal. Also, we study some invariants of Weyl type theorems. Moreover, we will prove that a class n-* paranormal operator is finite and it remains invariant under compact perturbation and some orthogonality results will be given.

Orthogonality Property $$\mathcal {O}$$ O and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.

Preconditioners in spectral approximation

Let H be a complex separable Hilbert space, and let A be a bounded self-adjoint operator on H. Consider the orthonormal basis B={e1,e2,…} and the projection Pn of H onto the finite-dimensional subspace spanned by the first n elements of B. The finite-dimensional truncations An=PnAPn shall be regarded as a sequence of finite matrices by restricting their domains to Pn(H) for each n. Many researchers used the sequence of eigenvalues of An to obtain information about the spectrum of A. But in many situations, these An’s need not be simple enough to make the computations easier. The natural question Can we use some simpler sequence of matrices Bn instead of An? is addressed in this article. The notion of preconditioners and their convergence in the sense of eigenvalue clustering are used to study the problem. The connection between preconditioners and compact perturbations of operators is identified here. The usage of preconditioners in the spectral gap prediction problems is also discussed. The examples of Toeplitz and block Toeplitz operators are considered as an application of these results. Finally, some future possibilities are discussed.

Spectral stability for compact perturbations of Toeplitz matrices

Let f be a regular real-valued non-constant symbol defined on the one dimensional torus T. Denote respectively by κ and T, its set of critical points and the associated Toeplitz matrix on l2(N). If V is a suitable compact perturbation, we prove that the operator T+V has no singular continuous spectrum and only finite point spectrum away from the set of thresholds f(κ). We also obtain some propagation estimates and apply these results to concrete examples.

Locking-Free Finite Element Methods for Poroelasticity

We propose a new formulation along with a family of finite element schemes for the approximation of the interaction between fluid motion and linear mechanical response of a porous medium, known as Biot's consolidation problem. The steady-state version of the system is recast in terms of displacement, pressure, and volumetric stress, and both continuous and discrete formulations are analyzed as compact perturbations of invertible problems employing a Fredholm argument. In particular, the error estimates are derived independently of the Lame constants. Numerical results indicate the satisfactory performance and competitive accuracy of the introduced methods.

On Well-Conditioned Spectral Collocation and Spectral Methods by the Integral Reformulation

Well-conditioned spectral collocation and spectral methods have recently been proposed to solve differential equations. In this paper, we revisit the well-conditioned spectral collocation methods proposed in [T.~A. Driscoll, {\it J. Comput. Phys.}, 229 (2010), pp.~5980-5998] and [L.-L. Wang, M.~D. Samson, and X.~Zhao, {\it SIAM J. Sci. Comput.}, 36 (2014), pp.~A907--A929], and the ultraspherical spectral method proposed in [S.~Olver and A.~Townsend, {\it SIAM Rev.}, 55 (2013), pp.~462--489] for an $m$th-order ordinary differential equation from the viewpoint of the integral reformulation. Moreover, we propose a Chebyshev spectral method for the integral reformulation. The well-conditioning of these methods is obvious by noting that the resulting linear operator is a compact perturbation of the identity. The adaptive QR approach for the ultraspherical spectral method still applies to the almost-banded infinite-dimensional system arising in the Chebyshev spectral method for the integral reformulation. Numerical examples are given to confirm the well-conditioning of the Chebyshev spectral method.

On the Jost Solution and Its Properties of Quantum Difference Equations with Hyperbolic Eigenparameter

This work is devoted to get Jost solution by the method of variation of parameters and properties of this Jost solution of a boundary value problem (BVP) depending on an hyperbolic eigenparameter. The results obtained are then used to find the Green function, resolvent and continuous spectrum of this BVP by using the methods of operator theory, methods of functional analysis and the general theory of difference equations. Moreover, using the Weyl compact perturbation theorem, we get that the operator L generated by the q -difference expression has continuous spectrum filling the segment [−2√ q , 2√ q ].

A junction condition by specified homogenization and application to traffic lights

Given a coercive Hamiltonian which is quasi-convex with respect to the gradient variable and periodic with respect to time and space at least "far away from the origin", we consider the solution of the Cauchy problem of the corresponding Hamilton-Jacobi equation posed on the real line. Compact perturbations of coercive periodic quasi-convex Hamiltonians enter into this framework for example. We prove that the rescaled solution converges towards the solution of the expected effective Hamilton-Jacobi equation, but whose "flux" at the origin is "limited" in a sense made precise by the authors in \cite{im}. In other words, the homogenization of such a Hamilton-Jacobi equation yields to supplement the expected homogenized Hamilton-Jacobi equation with a junction condition at the single discontinuous point of the effective Hamiltonian. We also illustrate possible applications of such a result by deriving, for a traffic flow problem, the effective flux limiter generated by the presence of a finite number of traffic lights on an ideal road. We also provide meaningful qualitative properties of the effective limiter.

Extended Supersymmetric Structures in Gapped and Superconducting Graphene

In view of the many quantum field theoretical descriptions of graphene in 2 + 1 dimensions, we present another field theoretical feature of graphene, in the presence of defects. Particularly, we shall be interested in gapped graphene in the presence of a domain wall and also for superconducting graphene in the presence of a vortex. As we explicitly demonstrate, the gapped graphene electrons that are localized on the domain wall are associated with four N = 2 one-dimensional supersymmetries, with each pair combining to form an extended N = 4 supersymmetry with non-trivial topological charges. The case of superconducting graphene is more involved, with the electrons localized on the vortex being associated with n one-dimensional supersymmetries, which in turn combine to form an N = 2n extended supersymmetry with non-trivial topological charges. As we shall prove, all supersymmetries are unbroken, a feature closely related to the number of the localized fermions and also to the exact form of the associated operators. In addition, the corresponding Witten index is invariant under compact and odd perturbations.

Large Isoperimetric Regions in Asymptotically Hyperbolic Manifolds

We show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds. Furthermore, we show that large coordinate spheres in compact perturbations of Schwarzschild-anti-deSitter are uniquely isoperimetric. This is relevant in the context of the asymptotically hyperbolic Penrose inequality. Our results require that the scalar curvature of the metric satisfies $R_{g}\geq -6$, and we construct an example of a compact perturbation of Schwarzschild-anti-deSitter without $R_{g}\geq -6$ so that large centered coordinate spheres are not isoperimetric. The necessity of scalar curvature bounds is in contrast with the analogous uniqueness result proven by Bray for compact perturbations of Schwarzschild, where no such scalar curvature assumption is required. This demonstrates that from the point of view of the isoperimetric problem, mass behaves quite differently in the asymptotically hyperbolic setting compared to the asymptotically flat setting. In particular, in the asymptotically hyperbolic setting, there is an additional quantity, the "renormalized volume," which has a strong effect on the large-scale geometry of volume.

On the degree for oriented quasi-Fredholm maps: its uniqueness and its effective extension of the Leray-Schauder degree

In a previous paper, the first and third author developed a~degree theory for oriented locally compact perturbations of $C\sp{1}$ Fredholm maps of index zero between real Banach spaces. In the spirit of a celebrated Amann--Weiss paper, we prove that this degree is unique if it is assumed to satisfy three axioms: Normalization, Additivity and Homotopy invariance. Taking into account that any compact vector field has a canonical orientation, from our uniqueness result we shall deduce that the above degree provides an effective extension of the Leray--Schauder degree.

A well-conditioned boundary integral equation for transmission problems of electromagnetism

We propose a new well-conditioned boundary integral equation to solve transmission problems of electromagnetism. This equation is well posed and appears as a compact perturbation of the identity leading to fast iterative solutions without the help of any preconditioner. Some numerical experiments confirm this result.

Stability of essential spectra of self-adjoint subspaces under compact perturbations

This paper studies stability of essential spectra of self-adjoint subspaces (i.e., self-adjoint linear relations) under finite rank and compact perturbations in Hilbert spaces. Relationships between compact perturbation of closed subspaces and relatively compact perturbation of their operator parts are first established. This gives a characterization of compact perturbation in terms of difference between the operator parts of perturbed and unperturbed subspaces. It is shown that a self-adjoint subspace is still self-adjoint under either relatively bounded perturbation with relative bound less than one or relatively compact perturbation or compact perturbation with a certain additional condition. By using these results, invariance of essential spectra of self-adjoint subspaces is proved under relatively compact and compact perturbations, separately. As a special case, finite rank perturbation is discussed. The results obtained in this paper generalize the corresponding results for self-adjoint operators to self-adjoint subspaces.

Two classes of operators with irreducibility and the small and compact perturbations of them

This paper gives the concepts of finite dimensional irreducible operators ((FDI) operators) and infinite dimensional irreducible operators ((IDI) operators). Discusses the relationships of (FDI) operators, (IDI) operators and strongly irreducible operators ((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an (FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in (ΣFDI)(X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in (ΣFDI)(X) with respect to the norm topology on a Banach space X with a Schauder basis, where (ΣFDI)(X):= {T ∈ B(X): T = Σi=1k ⊕Ti, Ti ∈ (FDI), k ∈ ℕ}.

On compact perturbation of two-particle Schrödinger operator on a lattice

We consider a system of two arbitrary quantum particles on three-dimensional lattice with certain dispersion functions (describing the transfer of the particle from one node to another) interacting by the attractive potential only on the nearest neighboring nodes. We find a class of potentials such that under perturbation of the two-particle operator h(k), corresponding to a two-particle system, with a potential from this class, the discrete spectrum of h(k) is preserved.

Gupta–Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space-Times

We give a complete framework for the Gupta-Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebras in the so-called Gupta-Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero modes. The only requirement, the absence of zero-resonance states, is shown to be stable under compact perturbations of topology and metric. Usual deformation arguments based on the time-slice axiom then lead to a construction of Gupta-Bleuler representations on a large class of globally hyperbolic space-times. As usual, the field algebra is represented on an indefinite inner product space, in which the physical states form a positive semi-definite subspace. Gauge transformations are incorporated in such a way that the field can be coupled perturbatively to a Dirac field. Our approach does not require any topological restrictions on the underlying space-time.

Hyperinvariant subspaces for some compact perturbations of multiplication operators

In this paper, a sufficient condition for the existence of hyperinvariant subspace of compact perturbations of multiplication operators on some Banach spaces is presented. An interpretation of this result for compact perturbations of normal and diagonal operators on Hilbert space is also discussed. An improvement of a result of [FX12] for compact perturbations of diagonal operators is also obtained.

Oriented Degree of Fredholm Maps: Finite-Dimensional Reduction Method

The complete oriented degree theory for proper nonlinear Fredholm zero-index maps and their compact perturbations is constructed by means of a finite-dimensional reduction method. Applications to solvability and bifurcations of boundary-value problems for nonlinear elliptic equations are provided.

Spectral points of type π_+ and type π_- of closed operators in indefinite inner product spaces

We introduce the notion of spectral points of type π+ and type π− of closed operators A in a Hilbert space which is equipped with an indefinite inner product. It is shown that these points are stable under compact perturbations. In the second part of the paper we assume that A is symmetric with respect to the indefinite inner product and prove that the growth of the resolvent of A is of finite order in a neighborhood of a real spectral point of type π+ or π− which is not in the interior of the spectrum of A. Finally, we prove that there exists a local spectral function on intervals of type π+ or π−.