International Journal of Pure and Applied Mathematics ————————————————————————– Volume 22 No. 3 2005, 343-359 THE SECOND REGULARIZED TRACE OF A SECOND ORDER DIFFERENTIAL OPERATOR WITH UNBOUNDED OPERATOR COEFFICIENT Ehliman Adiguzelov 1 , Pinar Kanar 2 § 1,2 Department of Mathematics Faculty of Science and Art Yıldız Technical University Davutpa¸ sa Campus, 34210, Esenler, Istanbul, TURKEY 1 e-mail: ehliman@yildiz.edu.tr 2 e-mail: pinarkanar@yahoo.com Abstract: In this work, a formula for the second regularized trace of second order differential operator with unbounded operator coefficient is found. AMS Subject Classification: 34L05, 47A10 Key Words: Hilbert space, self-adjoint, normal operator, resolvent, spectrum 1. Introduction Let H be a infinite dimensional separable Hilbert space. We denote the inner product in H by (.,.) and the norm in H by ‖.‖. Let H 1 = L 2 (H ; [0,π]) denote the set of all functions f from [0,π] into H which are strongly measurable and satisfy the condition  π 0 ‖f (x)‖ 2 dx< ∞. If the inner product of arbitrary two elements f and g of the space H 1 is defined as (f,g) H 1 =  π 0 (f (x),g(x))dx, then H 1 becomes a infinite dimensional separable Hilbert space [13]. The norm in the space H 1 is denoted by ‖.‖ 1 . σ ∞ (H ) denotes the set of all compact Received: June 13, 2005 c  2005, Academic Publications Ltd. § Correspondence author