On the Uniform Poincaré Inequality

We give a proof of the Poincaré inequality in W ^{1, p} (Ω) with a constant that is independent of Ω ? , where  is a set of uniformly bounded and uniformly Lipschitz domains in ? ⁿ . As a byproduct, we obtain the following: The first non vanishing eigenvalues λ₂(Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.     

On multicyclic operators and the Vasjunin-Nikol'ski206-1206-1206-1discotheca

If A is a bounded linear multicyclic operator acting on a complex Banach spaceX, then thedisc of A is defined by: disc A = sup(R ∈ Cyc A) min{dimR′: R′ ? R, R′ ∈ Cyc A}, where Cyc A denotes the family of all finite dimensional subspacesR ofX such that X = (R+AR+A ² R+?)^?. It is shown that if the set {λ ∈ ?: dim ker (λ-A)^* ≥ n} has nonempty interior (in particular, if A is a Fredholm operator of index -n), then disc A ≥ n+1. This result affirmatively answers a question of V.I. Vasjunin and N.K. Nikol'skiï. In the case whenX is a Hilbert space, it is shown that the set of all operators A such that A is n-multicyclic, but disc A =∞, is dense in the set of all n-multicyclic operators. If M_λ = "multiplication by λ" acting on the disk algebra (and many other spaces of continuous and/or analytic functions), then M_λ is cyclic, but disc M_λ = ∞. However, the analogous result is false if the disk algebra is replaced by the algebra of functions analytic on the disk and smooth on the boundary, or algebras of Lipschitz functions. If T is a multicyclic unicellular operator, then T is cyclic and disc T=1.     

Every completely polynomially bounded operator is similar to a contraction

It is proved that a bounded operator on a Hilbert space is similar to a contraction if and only if it is completely polynomially bounded. This gives a partial answer to Problem 6 of Halmos (Bull. Amer. Math. Soc.76 (1970). 877–933). The set of completely bounded maps between C¹-algebras is studied to obtain some structure, representation, and extension theorems for this class of maps. These allow a characterization of the completely bounded representations, on a Hilbert space, of any subalgebra of a C¹-algebra to be obtained. The result in the title follows by applying this characterization to the disk algebra.     

Compact operators without extended eigenvalues

A complex number λ is called an extended eigenvalue of a bounded linear operator T on a Banach space B if there exists a non-zero bounded linear operator X acting on B such that XT=λTX. We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set {1}.     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω?C, and in Fock spacesA ²(C ^N),N≧1.     

Lipschitz spaces and Suslin sets

Every bounded Suslin set in the plane is the set of eigenvalues of some bounded operator in a separable B-space.     

Inequalities for lower-order eigenvalues of a fourth-order elliptic operator

In this paper, we investigate the Dirichlet weighted eigenvalues problem of a fourth-order elliptic operator with variable coefficients on a bounded domain with smooth boundary in ? ⁿ . We establish some inequalities for lower-order eigenvalues of this problem. In particular, our results contain an inequality for eigenvalues of the biharmonic operator derived by Cheng, Huang, and Wei.     

Asymptotic analysis of a spectral problem in a periodic thick junction of type 3:2:1

Convergence theorems and asymptotic estimates (as ϵ→0) are proved for eigenvalues and eigenfunctions of a mixed boundary value problem for the Laplace operator in a junction Ω_ϵ of a domain Ω₀ and a large number N² of ϵ‐periodically situated thin cylinders with thickness of order ϵ=O(N⁻¹). We construct an extension operator that is only asymptotically bounded in ϵ on the eigenfunctions in the Sobolev space H¹. An approach based on the asymptotic theory of elliptic problem in singularly perturbed domains is used. Copyright © 2000 John Wiley & Sons, Ltd.     

On an integral equation approach for the exterior Robin problem for the Helmholtz equation

A boundary integral equation for the exterior Robin problem for Helmholtz's equation is analyzed in this paper. This integral operator is not compact. A proof based on a suitable regularization of this integral operator and the Fredholm alternative for the regularized compact operator was given by other authors. In this paper, we will give a direct existence and uniqueness proof for the boundary non-compact integral equation in the space settings C^1,λ(S) and C^0,λ(S), where S is a closed bounded smooth surface.     

On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

The conjugation problem on an infinite set of intervals

It is known that if path of integration consists of a finite number of intervals, then: (1) in the case of a Fredholm-type kernel, the index of the Fredholm operator is zero; (2) in the case of a Cauchy-type kernel, the index of the singular integration operator is a finite number (possible zero). Study of the conjugate boundary-value problem on an infinite set of intervals brings out new facts. The following may be noted: (1) A homogeneous boundary-value problem is always solvable in the classK, which is a natural generalization of that of piecewise analytic functions [1]. (2) Associated (conjugated) homogeneous boundary-value problems have any number of linearly independent solutions in the associated (conjugated) classes, so that the notion of class index is no longer relevant. (3) Associated (conjugated) homogeneous singular integral equations have any number of linearly independent solutions in the associated (conjugated) spacesL _p, L_q, p^?1+q^?1=1, so that the notion of operator index is no longer relevant The general theory of the problems under consideration is satisfactorily illustrated by the simplest case—a set of intervals on the real axis. For this reason the line of discontinuities (integration path) in the present paper is part of the real axis. The paper generalizes the results of [2–4]. Relevant work includes [5].     

Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

The spectral problem in a bounded domain Ω?Rⁿ is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ _j ⁰ } _j=1 ^∞ and {λ _j ^∞ } _j=1 ^∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated.     

Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators

In this paper, we prove that, if the product A=A₁?An is a Fredholm operator where the ascent and descent of A are finite, then Aj is a Fredholm operator of index zero for all j, 1?j?n, where A₁,…,An be a symmetric family of bounded operators. Next, we investigate a useful stability result for the Rako?evi?/Schmoeger essential spectra. Moreover, we show that some components of the Fredholm domains of bounded linear operators on a Banach space remain invariant under additive perturbations belonging to broad classes of operators A such as γ(Am)<1 where γ(⋅) is a measure of noncompactness. We also discuss the impact of these results on the behavior of the Rako?evi?/Schmoeger essential spectra. Further, we apply these latter results to investigate the Rako?evi?/Schmoeger essential spectra for singular neutron transport equations in bounded geometries.     

Bounds for ratios of the membrane eigenvalues

For a bounded planar region in R², we obtain the ratios of lower order eigenvalues of Laplace operator. Combining our results with the recursive formula in Cheng and Yang (2007) [11], we can obtain better upper bound of the (k+1)-th (k?3) membrane eigenvalues.     

Perturbation of Fredholm eigenvalues of linear operators

We use the reduction method, which allows one to reduce the study of perturbations of multiple eigenvalues to perturbations of simple eigenvalues, to analyze the general perturbation problem for Fredholm points of the discrete spectrum of linear operator functions analytically depending on the spectral parameter. The same method is used to study a perturbation of multiple Fredholm points of the discrete Schmidt spectrum (s-numbers) of a linear operator. We present an example of a problem on a perturbation of the domain of the Sturm–Liouville problem for a second-order differential operator.     

Cauchy problem for a degenerate heat equation in Banach spaces

We study the solvability of a degenerate heat equation with closed linear operators B multiplying the time derivative and A multiplying the Laplace operator in the class of generalized functions in Banach spaces. Under various assumptions on the operator pencil λB-A (it can be Fredholm of index zero, Fredholm, spectrally bounded, sectorial, or radial), we construct the fundamental operator function for the differential operator Bδ′(t) × δ ( $ \bar x $ ) ? Aδ(t) × Δδ ( $ \bar x $ ) and use it for the closed-form construction of the desired generalized solution of the Cauchy problem for the equation in question. We single out uniqueness classes for these solutions and analyze the relationship between continuous and generalized solutions.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.