Variational principles for real eigenvalues of self-adjoint operator pencils (original) (raw)

On the spectrum of linear operator pencils

Matematičnì studìï, 2019

We consider a linear operator pencil L(λ) = A − λB, λ ∈ C, where A and B are bounded operators on Hilbert space. The purpose of this paper is to study the conditions under which the spectrum of L(.) is the whole complex plane or empty. This leads to some criteria for the spectrum to be bounded.

On an eigenvalue problem of Ahmad and Lazer for ordinary differential equations

Proceedings of the American Mathematical Society, 1987

Lazer, we show the existence of a class of nonselfadjoint eigenvalue problems related to the equation y(n) + Xp(x)y = 0 for which the general eigenvalues comparison is not true. We use a comparison principle for the zeros of the corresponding Cauchy problem. This paper provides a contribution to the understanding of a problem raised by S. Ahmad and A. C. Lazer [1] in connection with the comparison of the eigenvalues for some multi-point boundary value problems which are not selfadjoint. One is given the equation (1) Lny + Xp(x)y = 0, where p(x) is a continuous function of constant sign on an interval /, A is a parameter, and Lny is a linear differential disconjugate operator of order n, that is, the only solution of Lny = 0 with n zeros on I (counting multiplicity) is y = 0. Let us consider the eigenvalue problem given by equation (1) and the system of boundary conditions ,, Lzy(a)=0, iG{ii,...,ik}, L]V(b)=0, JGiJu.-.Jn-k}, where o, b G I, 1 < k < n-1, Liy, i = 0,..., n-1, are the quasi-derivatives of y(x) (see [7]), and {t'i,..., ¿fc}, {ji, ■ ■ ■ ,jn-k) are two arbitrary sets of indices from the set {0,... ,n-1}. Problems of this type have been studied extensively (cf. [2, 3, 5]). In particular, Elias [5] has shown that if (-l)n_fcp(x) < 0, then the eigenvalues of problems (1) and (2) are real and nonnegative and form a divergence sequence {Am}m£N-Ahmad and Lazer [1] have considered a particular type of boundary condition (2), that is (3) y(a)=y'(a) =-= yik-1\a) = 0, y(b)=y'(b) =-=y(n-k-i\b)=0, and showed that if we set p = Pi, where p¿, i-1,2, are two continuous functions, considering the corresponding sequence of eigenvalues (A¿,m)m6N, i = 1,2, ordered by magnitude, then the condition (4)_ (-l)n-kp2(x) < (-l)"-fepi(x) < 0

Spectral properties of zeroth-order pseudodifferential operators

Journal of Functional Analysis, 1983

Let Q be a self-adjoint, classical, zeroth order pseudodifferential operator on a compact manifold X with a fixed smooth measure dx. We use microlocal techniques to study the spectrum and spectral family, (Es)sEIR, of Q as a bounded operator on L '(X, dx). Using theorems of Weyl (Rend. Circ. Mat. Palermo, 27 (1909), 373-392) and Kato ("Perturbation Theory for Linear Operators," Springer-Verlag, 1976) on spectra of perturbed operators we observe that the essential spectrum and the absolutely continuous spectrum of Q are determined by a finite number of terms in the symbol expansion. In particular Specs,, Q = range@@, 5)) where q is the principal symbol of Q. Turning the attention to the spectral family (ES/,, R, it is shown that if dE/ds is considered as a distribution on IR x XX X it is in fact a Lagrangian distribution near the set (u = 01 c T*(R XX x X)\O, where (s, x, y, o, <, a) are coordinates on T*(lR X XX X) induced by the coordinates (s, x, y) on [R x XX X. This leads to an easy proof thatf(Q) is a pseudodifferential operator iffE C"'(lR) and to some results on the microlocal character of E,. Finally, a look at the wavefront set of dE/ds leads to a conjecture about the existence of absolutely continuous spectrum in terms of a condition on q(x, <).

On the Spectra of Left-Definite Operators

Complex Analysis and Operator Theory, 2013

If A is a self-adjoint operator that is bounded below in a Hilbert space H; Littlejohn and Wellman showed that, for each r > 0; there exists a unique Hilbert space Hr and a unique self-adjoint operator Ar in Hr satisfying certain conditions dependent on H and A: The space Hr and the operator Ar are called, respectively, the r th left-de…nite space and r th left-de…nite operator associated with (H; A): In this paper, we show that the operators A; Ar; and As (r; s > 0) are isometrically isomorphically equivalent and that the spaces H; Hr; and Hs (r; s > 0) are isometrically isomorphic. These results are then used to reproduce the left-de…nite spaces and left-de…nite operators: Furthermore, we will see that our new results imply that the spectra of A and Ar are equal, giving us another proof of this phenomenon that was …rst established in . words and phrases. right-de…nite operator, left-de…nite operator, left-de…nite space, Hilbert scale, Sobolev space, self-adjoint operator, isometric isomorphism, similarity transformation, point spectrum, continuous spectrum. 1 2 LANCE L. LITTLEJOHN AND RICHARD WELLMAN

Variational analysis of an extended eigenvalue problem

Linear Algebra and its Applications, 1995

is known to arise in optimality conditions for the mathematical programming problem P(a) given by max{rctBx-2atx : ztz = l}, as well as in extended Rayleigh-Ritz type results pertaining to the one parameter *Research supported by Natural Sciences Engineering Research Council Canada operating grant A4641.