Compact perturbation of definite type spectra of self-adjoint quadratic operator pencils

Self-adjoint quadratic operator pencilsL()= ² A + B + C with a noninvertible leading operatorA are considered. In particular, a characterization of the spectral points of positive and of negative type ofL is given, and their behavior under a compact perturbation is studied. These results are applied to a pencil arising in magnetohydrodynamics.     

Banach algebras associated with linear operator pencils

A direct relationship between the theory of pseudoresolvents and the spectral theory of linear operator pencils is established.     

Self-adjoint quadratic operator pencils and elliptic problems

Functional Analysis and Its Applications -     

The linear factorization of polynomial operator pencils

Sufficient conditions are found for the linear factorization of polynomial operator pencils of arbitrary order in a Banach space. This factorization is generated by the solution of an appropriate operator equation.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 551–559, April, 1973.The author wishes to express his thanks to A. G. Kostyuchenko for the formulation of the problem.     

On the spectrum of a weak class of operator pencils of waveguide type

The paper introduces a new class of two parameter non‐overdamped operator pencils arising from evolution equations. We investigate spectral properties, including variational principles for “interior” points of the spectrum. Examples leading to pencils of the new class are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary value problems for a class of elliptic operator pencils

In this paper operator pencilsA(x, D, ) are studied which act on a manifold with boundary and satisfy the condition of N-ellipticity with parameter, a generalization of the notion of ellipticity with parameter as introduced by Agmon and Agranovich-Vishik. Sobolev spaces corresponding to the Newton polygon are defined and investigated; in particular it is possible to describe their trace spaces. With respect to these spaces, an a priori estimate is proved for the Dirichlet boundary value problem connected with an N-elliptic pencil.Supported in part by the Deutsche Forschungsgemeinschaft and by Russian Foundation of Fundamental Research, Grant 00-01-00387.     

On a certain class of linear Hamiltonian maps

We discovered a certain class of linear Hamiltonian maps which are defined by explicit time dependent Hamiltonian functions. Our method is the analogy of Moser's construction in 1986, although it is not so difficult, but we think the results are new.

     

On elliptic operator pencils with general boundary conditions

In this paper parameter-dependent partial differential operators are investigated which satisfy the condition of N-ellipticity with parameter, an ellipticity condition formulated with the use of the Newton polygon. For boundary value problems with general boundary operators we define N-ellipticity including an analogue of the Shapiro-Lopatinskii condition. It is show that the boundary value problem is N-elliptic if and only if an a priori estimate with respect to certain parameter-dependent norms holds. These results are closely connected with singular perturbation theory and lead to uniform estimates, for problems of Vishik-Lyusternik type containing a small parameter.Supported in part by the Deutsche Forschungsgemeinschaft and by Russian Foundation of Fundamental Research, Grant 00-01-00387.     

On the class number of certain imaginary quadratic fields

Theorem. Let 2$"> denote an integer, the square-free part of and the class number of the field . Then except for the case , divides .

     

On perturbations of matrix pencils with real spectra, a revisit

This paper continues earlier studies by Bhatia and Li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra. A unifying framework for creating crucial perturbation equations is developed. With the help of a recent result on generalized commutators involving unitary matrices, new and much sharper bounds are obtained.

     

On a certain class of commuting systems of linear operators

In this paper we describe the class of commuting pairs of bounded linear operators {A ₁,A ₂} acting on a Hilbert space H which are unitarily equivalent to the system of integrations over independent variables $$ (\tilde A_1 f)(x,y) = i\int_x^a {f(t,y)} dt, (\tilde A_2 f)(x,y) = i\int_y^b {f(x,s)} ds $$ in $ L_{\Omega _L }^2 $ , where ?? _L is the compact set in ? ₊ ² bounded by the lines x = a and y = b and by a decreasing smooth curve L = {((x, p(x)): p(x) ?? C _[0,a] ¹ , p(0) = b, p(a) = 0}.     

Remainder estimates in spectral asymptotic formulas for linear operator pencils

One gives a qualified remainder estimate in the asymptotic formulas of the distribution functions of the positive and negative spectra of a linear pencil of differential operators, where the lowest operator is not assumed to be elliptic.Translated from Problemy Maternaticheskogo Analiza, No. 9, pp. 34–56, 1984.     

On a certain class of operator equations with small parameter and regularization of ill-posed problems

The research was financially supported by the International Science Foundation (Grant NMW000).     

An algorithm for solving the eigenvalue problem for a class of operator pencils

We study the generalized eigenvalue problem for a uniformly hyperbolic or totally hyperbolic polynomial operator pencil of arbitrary order. In the context of the variational approach to the problem of determining the eigenvalues of such pencils we propose and justify a recursive algorithm for finding an eigenpair. To accelerate the convergence we propose using the Rayleigh-Ritz procedure. We give data from a numerical implementation of the algorithm. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 24–28.