Self-adjoint Linear Operator Research Articles

Spectrum of a particular bounded self-adjoint linear operator

Let be a connected bounded open set in R N , N 2, with lipschitzian boundary. The best constant in the Poincar e type inequality: j uj 2 C()k grad(u)k 2 1 ;8u2 L 2 ()=R is the inverse of the smallest spectral value of the the bounded self-adjoint linear operator T = div( ) 1 grad in L 2 ()=R ([4]). In this paper we show that, in the case of an elliptical domain of R 2 , the point spectrum of this operator is the set p(T )= fn; e n; 1; n2 N*g, where 1 is an eigenvalue

Operator Estimates for Fredholm Modules

AbstractWe study estimates of the typewhere φ(t) = t(1 + t2)−1/2, D0 = D0* is an unbounded linear operator affiliated with a semifinite von Neumann algebra , D − D0 is a bounded self-adjoint linear operator from and , where E(, τ) is a symmetric operator space associated with . In particular, we prove that φ(D) − φ(D0) belongs to the non-commutative Lp-space for some p ∈ (1,∞), provided belongs to the noncommutative weak Lr-space for some r ∈ [1, p). In the case and 1 ≤ p ≤ 2, we show that this result continues to hold under the weaker assumption . This may be regarded as an odd counterpart of A. Connes’ result for the case of even Fredholm modules.

On Unbounded p-Summable Fredholm Modules

We prove that odd unbounded p-summable Fredholm modules are also bounded p-summable Fredholm modules (this is the odd counterpart of a result of A. Connes for the case of even Fredholm modules). The approach we use is via estimates of the form ‖φ(D)−φ(D0)‖Lp(M,τ)⩽C·‖D−D0‖1/2, where φ(t)=t(1+t2)−1/2, D0=D*0 is an unbounded linear operator affiliated with a semifinite von Neumann algebra M, D−D0 is a bounded self-adjoint linear operator from M and (1+D20)−1/2∈Lp(M,τ), where Lp(M,τ) is a non-commutative Lp-space associated with M. It follows from our results that if p∈(1,∞), then φ(D)−φ(D0) belongs to the space Lp(M,τ).

Domain decomposition methods for eigenvalue problems

This paper proposes several domain decomposition methods to compute the smallest eigenvalue of linear self-adjoint partial differential operators. Let us be given a partial differential operator on a domain which consists of two nonoverlapping subdomains. Suppose a fast eigenvalue solver is available for each of these subdomains but not for the union of them. The first proposed method is a scheme which determines an appropriate boundary condition at the interface separating the two regions. This boundary condition can be derived from the zero of a nonlinear operator which plays the same role for the eigenvalue problem as the Steklov–Poincaré operator for the linear equation. An iterative method can be used to solve this nonlinear equation, yielding the exact boundary condition at the interface. This enables the determination of the eigenpair on the whole domain. The same concept can be applied to a Schwarz alternating method for the eigenvalue problem in case the subdomains overlap. A nonoverlapping scheme (in the spirit of Schur complement) will also be discussed. These ideas are also applicable to a domain imbedding algorithm.

Constrained approximate controllability

In the paper, constrained approximate controllability for linear dynamical systems described by abstract differential equations with unbounded control operator is considered. Using methods of spectral analysis for linear self-adjoint operators and general constrained controllability results given by Son (1990), necessary and sufficient conditions of the constrained approximate controllability for the piecewise polynomial controls with values in a given cone are formulated and proved. Moreover, as illustrative examples, constrained approximate boundary controllability of one-dimensional distributed parameter dynamical systems described by partial differential equations of parabolic type with Dirichlet and Neumann boundary conditions are investigated. The constrained controllability conditions obtained in the paper represent an extension of the unconstrained controllability results given by Glothin (1974) and Klamka (1991).

Spectral characterization of solutions to systems of linear differential equations

A spectral characterization of solutions of abstract linear differential equation systems is given. The characterization is in terms of the spectrum of a related continuous self-adjoint linear operator.

Quasilinear degenerate parabolic equations of Kirchhoff type

We investigate the evolution problem u' + m(|A 1/2 u| 2 )Au = 0, u(0) = u 0 ∈ H, where H is a Hilbert space, A is a self-adjoint linear non-negative operator on H with domain D(A), and m:[0, +∞[→[0, +∞[ is a continuous function. We prove that if u 0 ∈ D(A 1/2 ), and m(|A 1/2 u 0 | 2 ) ¬= 0, then there exists at least one global solution, which is unique if either m never vanishes, or m is locally Lipschitz continuous. Moreover, we prove that if (1 + ρ)m(ρ) ≥ c > 0 for all ρ ≥ 0, then this problem is well posed in H. On the contrary, if for some α > 1 it happens that (1 + ρ α )m(ρ) ≤ c < + ∞ for all ρ ≥ 0, then this problem has no solution if u 0 ¬∈ D(A β ) with β small enough. We apply these results to degenerate parabolic PDEs with non-local non-linearities.

Exponential decay for distributed bilinear control systems with damping

Let Ω be a bounded open domain in ℝ N ,A an unbounded, selfadjoint, positive and coercive linear operator onL 2 (Ω). We consider feedback stabilization for the distributed bilinear control systemy″(t)+Ay(t)+Dy′(t)+u(t)By(t)=0, whereD andB are linear bounded operators fromL 2(Ω) toL 2(Ω). Under suitable assumptions onB andD, a nonlinear feedback ensuring uniform exponential decay of solutions is given. Various applications to vibrating processes are presented.

A Class of Pattern-Forming Models

A general class of nonlinear evolution equations is described, which support stable spatially oscillatory steady solutions. These equations are composed of an indefinite self-adjoint linear operator acting on the solution plus a nonlinear function, a typical example of the latter being a double-well potential. Thus a Lyapunov functional exists. The linear operator contains a parameter ρ which could be interpreted as a measure of the pattern-forming tendency for the equation. Examples in this class of equations are an integrodifferential equation studied by Goldstein, Muraki, and Petrich and others in an activator-inhibitor context, and a class of fourth-order parabolic PDE's appearing in the literature in various physical connections and investigated rigorously by Coleman, Leizarowitz, Marcus, Mizel, Peletier, Troy, Zaslavskii, and others. The former example reduces to the real Ginzburg-Landau equation when ρ = 0 .

Measurable enumeration of eigenelements

We prove that for eigenelements of a measurable family of self-adjoint linear operators in separable Hilbert space there exists a measurable enumeration. We also prove a similar result for measurable families of bounded linear operators having at most countably many eigenvalues (under certain restrictions on the parameter space). The proof of the latter result is based on descriptive set theory, while in the case of self-adjoint (and some more general) operators the proof is constructive.

Global Nonexistence for Abstract Evolution Equations with Positive Initial Energy

In this paper we consider the problem of noncontinuation of solutions of the initial value problem for abstract evolution equations of the formPutt+Q(t)ut+A(t,u)=F(t,u),t∈J=[0,∞),wherePandQare linear self-adjoint operators, andA(t,u) andF(t,u) are respectively a linear operator inu(typically of differential type) and a nonlinear driving force. Our principal concern is the noncontinuation (or blow-up) of solutions when the initial energy is positive, but appropriately bounded.

On Dirichlet's Principle and Its Applications

By means of Dirichlet's variational method and the theory of distributions we study the spectrum of a linear self-adjoint operator and give some applications to the theory of differential operators in unbounded domain. We also link this subject to the theory of numbers.

A numerical algorithm for solving the generalized eigenvalue problem for completely continuous self-adjoint operators with nonlinear spectral parameter

We construct and justify a numerical algorithm for finding the generalized eigenvalues and eigenfunctions for self-adjoint positive semidefinite linear operators with nonlinear spectral parameter. We give an example of its application to finding the branch points of solutions of a class of nonlinear integral equations of Hammerstein type.

Quadratic optimization of fixed points of nonexpansive mappings in hubert space

Finding an optimal point in the intersection of the fixed point sets of a family of nonexpansive mappings is a frequent problem in various areas of mathematical science and engineering. Let be nonexpansive mappings on a Hilbert space H, and let be a quadratic function defined by for all , where is a strongly positive bounded self-adjoint linear operator. Then, for each sequence of scalar parameters (λn) satisfying certain conditions, we propose an algorithm that generates a sequence converting strongly to a unique minimizer u* of Θ over the intersection of the fixed point sets of all the Ti’s. This generalizes some results of Halpern (1967), Lions (1977), Wittmann (1992), and Bauschke (1996). In particular, the minimization of Θ over the intersection of closed convex sets Ci can be handled by taking Ti to the metric projection onto Ci without introducing any special inner products that depends on A. We also propose an algorithm that generates a sequence converging to a unique minimizer of Θ over , where K is a given closed convex set and for positive weights . This is applicable to the inconsistent case as well.

Some new results on global nonexistence for abstract evolution with positive initial energy

where P and Q(t) are linear self-adjoint operators, and A(t, u) and F (t, u) are typically a divergence operator in u and a nonlinear driving force. Other versions of (1.1) were considered earlier by Levine [3–6], for which he introduced the important technique of “concavity” analysis of auxiliary second order differential inequalities. In all these papers the principal mechanism of blow–up was the assumption of negative initial energy. In an interesting paper [10], which has just appeared, Ono has also used concavity analysis to study blow–up, but in the more general case when the initial energy is allowed to take appropriately small positive values. His analysis primarily considers linear wave operators, and moreover is restricted to bounded domains in R. (It should, however, be added that Ono also allows Kirchhoff type operators, an added generalization but without serious affect on the principal ideas.) Here we discuss some extentions of Ono’s analysis to the abstract equation (1.1), see Theorem 1. Moreover, in concrete cases, we introduce appropriate methods to treat divergence structure operators in unbounded domains (including but not necessarily restricted to R). Our conclusions also yield a larger class of initial data than in [10] for which blow-up must occur; see Remark 1 in Section 3. In the next section we give a precise meaning to equation (1.1), and give our main abstract theorem. Section 3 discusses a divergence structure equation in R for which blow–up occurs for positive initial energy, even for unbounded domains. Here the primary new idea, in comparison with [7] and [10], is to introduce an appropriate coercive operator associated with the equation. Proofs of the results described here will appear in the forthcoming paper [13].

Jensen’s operator inequality for functions of two variables

The operator convex functions of two variables are characterized in terms of a non-commutative generalization of Jensen's inequality. 1. FUNCTIONAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES The tensor product of two square matrices A = (aij) and B of order n can be represented as the matrix allB ... alnB (1) 1@B anl B .*.** aB / which is of order n2. However, if A and B are block matrices ( A21 A22) B = B21 B22) of order 2n, then it is often more convenient to represent the tensor product A 0 B as the block matrix / All 0 Bil All 0 B12 A12 0 Bil A12 0) B12 (2) A11? B21 A11 0 B22 A12 0 B21 A12 0 B22 (2) A21 B11 A21 0 B12 A22 X B1i1 A22 $' B12 A21 0 B21 A21 0 B22 A22 0 B21 A22 0 B22 / The definition according to (2) is unitarily equivalent to the definition according to (1), and no confusion will occur as long as the two representations are not mixed. The latter representation has the benefit of rendering formulas for block matrices more transparent and will be used throughout this paper. A similar representation will be used for tensor products of block matrices of bounded linear operators on a Hilbert space. Koranyi [10] considered functional calculus for functions of two variables. Let f: I x J -* R be a function of two variables defined on the product of two intervals, and let A, B be selfadjoint linear operators with finite spectra on a Hilbert space. If the spectrum of A is contained in I, and the spectrum of B is contained in J, Received by the editors February 2, 1996. 1991 Mathematics Subject Classification. Primary 47A63; Secondary 47A80, 47Bxx. ( 1997 American Mathematical Society 2093 This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:17:33 UTC All use subject to http://about.jstor.org/terms

Constrained approximate boundary controllability

In this paper, constrained approximate boundary controllability for linear disturbed parameter dynamical systems described by partial differential equations of parabolic-type with mixed boundary conditions is considered. Using methods of spectral analysis for linear self-adjoint operators and general constrained controllability results, given by Son (1990), necessary and sufficient conditions for constrained approximate boundary controllability are formulated and proved. Moreover, in illustrative examples the constrained approximate boundary controllability of 1D distributed parameter dynamical systems is investigated. The constrained controllability conditions obtained represent an extension of the unconstrained controllability results given by Glothin (1974).

Inertial manifolds and normal hyperbolicity

Our aim in this article is to derive an existence theorem of inertial manifolds for fairly general equations with a self-adjoint or nonself-adjoint linear operator in a Banach space setting. A sharp form of the spectral gap condition is given. Many other properties are proven including an interesting characterization of the inertial manifold and the normal hyperbolicity of the inertial manifold.

The Approximation of the Optimal Control of Vibrations—A Geometrical Approach

We employ concepts from Banach space geometry in order to examine the problem of approximating the optimal distributed control of vibrating media whose motion is governed by a wave equation with a 2n-order self-adjoint and positive-definite linear differential operator. We show that this geometrical approach, arrived at via duality theory, provides the exact framework in which the approximation problem must be placed in order to get the correct convergence results, for it is here that the necessary and sufficient conditions for the approximate norm or time minimal control can be fully developed. Using the theory of Asplund, we are also able to improve the traditional weak * convergence results for the more difficult case of L∞ controls. Finally, we consider certain numerical examples which help illustrate our theoretical results.

Lax–Phillips theory for scattering by thin elastic bodies

The scheme of construction of linear self-adjoint operators for acoustic systems with an interaction of sound and vibration is proposed. Kinetic energy of a system induces the basic Hilbert space L2(Ω), where Ω denotes the domain filled with fluid. Potential energy of the system induces the quadratic form on this Hilbert space and, consequently, the positive operator that governs the evolution. Unlike earlier theories, our operators are perturbations of the Laplace operator in L2 spaces. The operators may also be obtained as extensions of symmetric operator (−Δ) defined on C∞0(Ω) into a wider Hilbert space. Functions from their domains have an additional smoothness of their derivatives on the boundary; the smoothness follows from the elasticity of the boundary. The scheme allows one to describe inhomogeneities of vibrating objects concentrated at a point or on a line (such as cracks and stiffeners). It is demonstrated that Lax–Phillips scattering theory may be generalized for scattering by thin elastic objects. The necessary condition for the existence of the incoming and the outgoing subspaces is derived in terms of spectral properties of the operator of the isolated elastic object.