Some properties of the KMS-Function

Abstract: Let, for j = 0, 1, H-j = H-J* greater than or equal to -w(j) > -infinity, be a self-adjoint operator in the Hilbert spaces H-j. Let T : H-1 --> H-0 be a linear operator with domain in H-1 and range in H-0. Let V-j(t) = exp(-tH(j)), t greater than or equal to 0, be the strongly continuous semigroup generated by Hj, j = 0, 1 If the operators (aI + H-0) V-0 (t(0)) T (aI + H-1)(-1) and (aI + H-0)(-1) T (aI + H-1) V-1 (t(0)) are compact, (Hillbert-Schmidt, Trace class), then so is the operator integral (t0)(0) V-0(u)TV1(T-0-u)du. The result is applicable if T = JH(1) - H(0)J, where J : H-1 --> H-0 is a bounded linear (identification) operator. In this case integral (t0)(0) V-0(u)TV1(t(0) - u)du = V-0(t(0)) J - JV(1) (t(0)); i.e. the difference of the semigroups. Some convergence and approximation results are presented as well. For example the operator integral (t0)(0) V-0(u)TV1(t(0) - u)du is expressed in terms of the operator t(0)V(0) (t(0)/2) TV1 (t(0)/2)