On the spectrum of a self-adjoint polynomial pencil

Abstract

517.984.46 1. We consider the polynomial operator pencil k--1 P(A) = AkE + ~ AJAj -Ao, D(P(A)) = D(Ao), j=l in a separable Hilbert space H, where the Aj (j = 1,..., k -1) are symmetric operators, A0 = A~ _> E, and D(Aj) = D(Ao) (j = 1,..., k-l). It is assumed that (j-v)A i >_ 0 and [Aju[ < ~lAoul+ K, lu I for any r > 0 and u E D(Ao), j = 1,..., k-l, where [-[ is the normin H, A'~ > 0,and vE {1,... ,k-l}.We study the position of the spectrum of the pencil P(A) in the complex plane. Some estimates for the operator functions P(X) and P-I(A) axe established. The results are applied to find the spectral asymptotics of a pencil of differential operators and a differential operator that depends polynomially on a small parameter.The book [1] and the article [2] contain a detailed bibliography on the theory of polynomial operator pencils. We also note the papers [3-6].The spectral theory of pencils of pseudodifferential operators and nonself-adjoint problems with paraxneters that axe elliptic in the Agmon-Douglis-Nirenberg sense was considered in Agranovich's recent papers [7-9]. We set ~p = min{~r/u, Tr/(k-u)}, 9 = {z e C : [argz[ < ~}, and w(A) = [A[/dist{X,R+U0~}(o # e c). Let the symbols (., 9 ) and [[ -[[ denote the inner product in H and the norm of a bounded operator in H, respectively. Theorem 1. The spectrum of the pencil P(A) lies on R+ and outside the sector ~ and the inequalities [[P-I(X)[ I _< Mw(A)IA] -k, [[A1/2P-'(A)[I <_ Mw(A)IA[ -k/2 (A e 9 \ R+), (Aou, u) <_ Mw(A)I(P(A)u, ~)1 (V~ e D(Ao), A e { \ R+) hold. Moreover, if Ao 1 e ~oo(g), then the number Y(t) of positive eigenvalues of the operator P(t), t > O, coincides with the number ./go(t) of positive eigenvalues of the pencil P(A) that do not ezceed t. The definitions of the functions N(t) and ~U0 (t) take into account the multiplicities of the corresponding eigenvalues. The proof of the relation g(t) = JV0(t) is based on Theorem 3 in [10].