Perturbation Theorem Research Articles

Perturbation theorems for local integrated semigroups and their applications

Perturbation theorems for local integrated semigroups and their applications

A perturbation theorem for linear Hamiltonian systems with bounded orbits

This paper concerns the Sacker-Sell spectral decomposition of aone-parametric perturbation of a non-autonomous linear Hamiltoniansystem with bounded solutions. Conditions ensuring the continuousvariation with respect to the parameter of the spectral intervalsand subbundles are established. These conditions depend on theperturbation direction and are closely related to the topologicalstructure of the flows induced by the initial system on the realand complex Lagrange bundles.

The [formula omitted] theory of divergence form elliptic operators under the Dirichlet condition

Let A be the 2 m th-order elliptic operator of divergence form with bounded measurable coefficients defined in a domain Ω of R n with smooth boundary. For 1 < p < ∞ we regard A as a bounded operator from the L p Sobolev space H 0 m , p ( Ω ) to H - m , p ( Ω ) . Trying to extend the result which has been obtained only in the case Ω = R n or in the case m = 1 to the general case, we succeed in constructing the resolvent ( A - λ ) - 1 and estimating its operator norm for some λ when the leading coefficients are uniformly continuous. Applying the result for L p resolvents, we show that the operator associated with A in L p ( Ω ) generates an analytic semigroup and obtain exponential decay estimates for the heat kernel, the resolvent kernel and their derivatives of order up to m - 1 . We also give a perturbation theorem for heat kernels.

Maximal Regularity for Nonautonomous Evolution Equations

Abstract We derive sufficient conditions, perturbation theorems in particular, for nonautonomous evolution equations to possess the property of maximal Lp regularity.

Théorème local pour chaînes de Markov de probabilité de transition quasi-compacte. Applications aux chaînes V-géométriquement ergodiques et aux modèles itératifs

Let Q be a transition probability on a measurable space E, let ( X n ) n be a Markov chain associated to Q, and let ξ be a real-valued measurable function on E such that ( ξ ( X n ) ) n ⩾ 0 satisfies a central limit theorem. Under functional hypotheses on the action of Q and its Fourier kernels Q ( t ) , we establish a local limit theorem. We use the spectral method of Nagaev improved by a perturbation theorem of Keller and Liverani. The conditions required here on Q ( t ) are weaker than the ones usually imposed when the standard perturbation theorem is used in the spectral method. We give some applications to V-geometric ergodic chains and to Lipschitz iterative models.

Asymptotic Behaviour of Parabolic Problems with Delays in the Highest Order Derivatives

We use semigroup methods to investigate the partial functional differential equation $u’(t)=Au(t)+ \int_{-r}^0 dB(\theta)u(t+\theta)$ for a sectorial operator $A$ on a Banach space $X$ and a function $B:[-r,0] \to\cL(D(A),X)$ of bounded variation having no mass at 0. Using a perturbation theorem due to Weiss and Staffans, we construct the solution semigroup on a product space in order to solve the delay equation in a classical sense. Employing the spectrum of the semigroup and its generator, we then study exponential dichotomy and stability of solutions. If $X$ is a Hilbert space, %C% these properties can be characterized by estimates on $(\la-A-\widehat{dB}(\la))^{-1}\in\cL(X,D(A))$. Related results on stability also hold for general Banach spaces. The case $B=\eta A$ with scalar valued $\eta$ is treated in some detail.

Perturbation theorems for upper and lower semi-Fredholm linear relations

Perturbation theorems for upper and lower semi-Fredholm linear relations

The Lp resolvents of second-order elliptic operators of divergence form under the Dirichlet condition

Let A be the 2 mth-order elliptic operator of divergence form with bounded measurable coefficients defined in a domain Ω of R n . For 1< p<∞ we regard A as a bounded linear operator from the L p Sobolev space H 0 m,p(Ω) to H −m,p(Ω) . It is known that when Ω= R n , we can construct the resolvent ( A− λ) −1 and estimate its operator norm for some λ if the leading coefficients are uniformly continuous. In this paper, we try to extend this result to a general domain. It is successful when m=1 if Ω is the half-space or a domain with C 2 bounded boundary. For m>1 it is shown that the problem is reduced to the case where Ω is the half-space and A is a homogeneous operator with constant coefficients. We also give a perturbation theorem.

Central limit theorems for iterated random Lipschitz mappings

Let M be a noncompact metric space in which every closed ball is compact, and let G be a semigroup of Lipschitz mappings of M. Denote by (Yn)n≥1 a sequence of independent G-valued, identically distributed random variables (r.v.’s), and by Z an M-valued r.v. which is independent of the r.v. Yn, n≥1. We consider the Markov chain (Zn)n≥0 with state space M which is defined recursively by Z0=Z and Zn+1=Yn+1Zn for n≥0. Let ξ be a real-valued function on G×M. The aim of this paper is to prove central limit theorems for the sequence of r.v.’s (ξ(Yn,Zn−1))n≥1. The main hypothesis is a condition of contraction in the mean for the action on M of the mappings Yn; we use a spectral method based on a quasi-compactness property of the transition probability of the chain mentioned above, and on a special perturbation theorem.

On the Essential Lower Bound of Elements in von Neumann Algebras

In this paper we obtain a number of interesting results on the connection that exists between the essential lower bound of an element in a von Neumann algebra and its Fredholm properties. These connections lead to an extension of a well-known perturbation theorem on Fredholm operators of index zero. Finally we obtain for special cases necessary and sufficient conditions for elements in the closure of the group of invertible elements to be essentially topological divisor of zero

Multiplicative perturbations of C-regularized resolvent families.

This paper presents two new theorems for multiplicative perturbations of C-regularized resolvent families, which generalize the previous related ones for the resolvent families.

Perturbation theorems for positive eigenvalues of countably condensing maps

Using a homotopy extension theorem, we give a perturbation theorem for countably condensing maps in a more general setting. Moreover, we prove perturbation theorems for positive eigenvalues of countably condensing maps in locally convex topological vector spaces which include the case of condensing maps due to Jerofsky [1].

On matrix perturbations with minimal leading Jordan structure

We show that any matrix perturbation of an n× n nilpotent complex matrix is similar to a matrix perturbation whose leading coefficient has minimal Jordan structure. Additionally, we derive the property that, for matrix perturbations with minimal leading Jordan structure, the sufficient conditions of Lidskii's perturbation theorem for eigenvalues are necessary too. It is further shown how minimality can be obtained by computing a similarity transform whose entries are polynomials of degree at most n. This relies on an extension of both Lidskii's theorem and its Newton diagram-based interpretation.

An extension of Henrici theorem for the joint approximate spectrum of commuting spectral operators

AbstractGiven two m-tuples of commuting spectral operators on a Hilbert space, T = (T1,…, Tm) and S = (S1,…, Sm), an extended version of Henrici perturbation theorem is obtained for the joint approximate spectrum of S under perturbation by T. We also derive an extended version of Bauer-Fike theorem for such tuples of operators. The method used involves Clifford algebra techniques introduced by McIntosh and Pryde.

Perturbations of Bi-continuous Semigroups with Applications to Transition Semigroups on C b (H)

We prove an unbounded perturbation theorem for bi-continuous semigroups on the space of bounded, continuous functions on the Hilbert space H. This is applied to the Ornstein-Uhlenbeck semigroup, thus providing a purely functional analytic approach to the existence of transition semigroups on Cb(H) with bounded non-linear drift.

Perturbation theorems for a-times integrated semigroups

We prove perturbation results for α-times integrated semigroups assuming relative “smallness” conditions for the perturbation B on a halfplane. If A is a semigroup generator on a uniformly convex Banach space, then these conditions on B already imply that A + B generates a once integrated semigroup. As an illustration we consider Schrodinger operators and higher order differential operators.

Error analysis of signal zeros: a projected companion matrix approach

An error analysis of so-called signal zeros of polynomials associated with exponentially damped/undamped signals is performed and zero error bounds are derived. The bounds are in terms of the angle between the exact and approximate signal subspace, the signal parameter themselves, the polynomial degree, and the error on the polynomial coeficients. The key idea behind the analysis is to regard signal zeros as eigenvalues of projected companion matrices and then to generate error bounds by exploiting perturbation theorem for eigenvalues. The conclusion drawn from the bounds is that the signal zeros become relatively insensitive to small perturbations on the polynomial coefficients as long as the polynomial degree is large enough and the zeros are extracted as eigenvalues of projected companion matrices. Also, the bounds suggests that signal zero estimates derived from projected companion matrices are more accurate than those obtained from the companion matrices themselves. Illustrative numerical results are provided.

A perturbation theorem for operator semigroups in Hilbert spaces

We prove a perturbation result for C0-semigroups on Hilbert spaces and use it to show that certain operators of the form Au = iu(2k) + V · u(l) on L2 (ℝ) generate a semigroup that is strongly continuous on (0, ∞).

Coexistence for a resource-based growth model with two resources

We investigate the coexistence of positive steady-state solutions to a parabolic system, which models a single species on two growth-limiting, non-reproducing resources in an un-stirred chemostat with diusion. We establish the existence of a positive steady-state solution for a range of the parameter (m; n), the bifurcation solutions and the stability of bifurcation solutions. The proof depends on the maximum principle, bifurcation theorem and perturbation theorem.

A generalization of frame perturbation in Hilbert subspaces and its application to wavelet subspaces

A perturbation theorem for frames in a Hilbert space is obtained, which is a generalization of a result by Christensen. Based on this general result, an irregular sampling theorem for frames in wavelet subspaces is established, which has a previous result as a special case.