K.regularized trace formula for the Sturm-Liouville equation with operator coefficient contained spectral parameter in the boundary condition

ABSTRACT Let H seperable Hubert space.In Hilbert space Hx = L2([ 0,1 \h)@ H define operator L generated by Sturm-Liouville differantial equation - y" + Q(x)y = Ay 0 < x < 1 and boundary conditions }'(0) = 0 -y(\) = *y'(l) Here Q(x) is a operator-function defined in [0,1] interval. Its values self-adjoint kernel operators in space H, and A, is a komplex spectral parameter. In this study for the L operator, k ( k > 2 ) is a any natural number which is given a regularized trace concept and obtained formula for k regularized trace. In this thesis, while Q(x) mentioned here is a reel valued skaler function and H=C, investigated first as a higher regularized trace formula. Key words: Weak derivative, analytic operator- function, essential spectrum, spectral expansion, regularized trace