Fifth International Conference of Applied Mathematics and Computing , University of Chemical Technology and Metallurgy, Plovdiv, Bulgaria, 12 - 18 August 2008, pp.11
Let L be operator which is formed by the differantial expression `(y) = (−1)^m.y^ (2m) (x) + Ay(x) + Q(x)y(x) in the space H1 = L2(0, π; H), with the boundary condition y^ (2i−1)(0) = y^ (2i−1)(π) = 0 , (i = 1, 2, ..., m) where H is an infinite dimensional separable Hilbert space. Here, A is an unbounded self adjoint operator in H and, for every x ∈ [0, π], Q(x) is a self-adjoint kernel operator in H. Assuming the operator A and the operator function Q(x) satisfy some additional conditions, it has been found a formula for the regularized trace of L. Here, n1 < n2 < ... and j1, j2, ... are sequences of natural numbers with a particular property. Furthermore, for Q(x) satisfying certain conditions
λ1 ≤ λ2 ≤ ... , and for Q(x) ≡ O
µ1 ≤ µ2 ≤ ... are the eigenvalues of the operators L, respectively; and
ϕ1, ϕ2, ... is a complete orthonormal sequence consisting of eigenvectors
of the operator A.