  We consider a perturbation determinant for pairs of nonpositive (in a sense
of Komatsu) operators on Banach space with nuclear difference and prove a
generalization of the important formula for the logarithmic derivative of this
determinant. To this end the Frechet differentiability of operator monotonic
(negative complete Bernstein) functions of negative and nonpositive operators
on Banach spaces is investigated. The results may be regarded as a contribution
to the Hirsch functional calculus.
