International Journal of Pure and Applied Mathematics ————————————————————————– Volume 63 No. 3 2010, 255-268 THE SUM OF SUBTRACTION OF THE EIGENVALUES OF TWO SELF ADJOINT DIFFERENTIAL OPERATORS WITH UNBOUNDED OPERATOR COEFFICIENT ¨ Ozlem Baksi 1 § , Yonca Sezer 2 , Serpil Karayel 3 1,2,3 Department of Mathematics Faculty of Arts and Sciences Yıldız Technical University Davutpas . a, Istanbul, 34210, TURKEY Abstract: In this work, a formula for the sum of the eigenvalues of two second order differential operators with unbounded operator coefficient is found. AMS Subject Classification: 47A10, 34L20 Key Words: Hilbert space, self adjoint operator, kernel operator, eigenvalues, spectrum 1. Introduction Let H be a separable Hilbert space. Let us consider the following differential expression in the space H 1 = L 2 (0,π; H ) ℓ 0 (y)= −y ′′ (x)+ Ay(x). Here, the operator A : D(A) → H in the space H satisfies the conditions A = A ∗ ≥ I A −1 ∈ σ ∞ (H ). Let γ 1 ≤ γ 2 ≤ ... ≤ γ n ≤ ... be the eigenvalues of the operator A and ϕ 1 ,ϕ 2 , ...ϕ n , ... be the orthonormal eigenvectors corresponding to these eigenval- ues. Here, each eigenvalue is added according to its own algebraic multiplicity number. Let D(L ′ 0 ) denote the set of the functions y(x) in the space H 1 satisfying the conditions: Received: August 19, 2009 c  2010 Academic Publications § Correspondence author