Variational principles for real eigenvalues of self-adjoint operator pencils in nonseparated root zones are studied. The main concern of this paper is the variational theoryof the spectrum for a class of self-adjoint operator pencils. | Turk J Math 31 (2007) , 43 – 52. ¨ ITAK ˙ c TUB Spectral Problems for Operator Pencils in Non-Separated Root Zones∗ M. Hasanov Abstract Variational principles for real eigenvalues of self-adjoint operator pencils in nonseparated root zones are studied. Key Words: Operator pencils, eigenvalues, variational principles. 1. Introduction Let L(λ) be a function defined on an interval [a, b] ⊂ R, whose values are operators in a Hilbert or Banach space. Such functions are called operator functions or operator pencils. Linear pencils of the form L(λ) = A − λB and polynomial pencils of the form L(λ) = λn An +λn−1 An−1 +.+λA1 +A0 , where A, B and Ai , i = 0, 1, ., n are operators, form important subclasses of operator pencils. In general, an operator pencil L(λ) may be analytic, smooth or nonsmooth. Polynomial pencils arise mainly from the evolution of equations in abstract spaces (see [10]) but nonpolynomial pencils arise from equations depending on a parameter. The main concern of this paper is the variational theory of the spectrum for a class of self-adjoint operator pencils. The spectrum of an operator pencil L(λ) is defined in the following way: We say that λ ∈ σ(L) if and only if 0 ∈ σ(L(λ)), where σ(L) denotes the spectrum of the operator pencil L(λ) and σ(L(λ)) denotes the spectrum of the operator L(λ) which 2000 AMS Mathematics Subject Classification: Primary 47A75; 47A56. Secondary 49R50; 34L15 ˙ of the author supported by NATO B2 program of TUBITAK. ∗ Research 43 HASANOV is the value of the operator pencil L(λ) at the point λ. The set of eigenvalues σe (L), the continuous spectrum σc (L) and other spectral sets are defined analogously. In particular, λ ∈ [a, b] is called an eigenvalue of the pencil L(λ) if there exists a vector x = 0, called an eigenvector such that L(λ)x = 0. Evidently, if L(λ) = A − λI then σ(L) = σ(A). It is well known that discrete eigenvalues of a self-adjoint operator A in a Hilbert space H, which lie below or above the .
