Perturbation determinants on Banach spaces and operator differentiability for Hirsch functional calculus

We consider a perturbation determinant for pairs of nonpositive (in a sense of Komatsu) operators on Banach space with nuclear difference and prove the 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.

Some Applications of the Perturbation Determinant in Finite von Neumann Algebras

In the finite von Neumann algebra setting, we introduce the concept of a perturbation determinant associated with a pair of self-adjoint elements H0 and H in the algebra and relate it to the concept of the de la Harpe-Skandalis homotopy invariant determinant associated with piecewise C1-paths of operators joining H0 and H. We obtain an analog of Krein's formula that relates the perturbation determinant and the spectral shift function and, based on this relation, we derive subsequently (i) the Birman-Solomyak formula for a general non-linear perturbation, (ii) a universality of a spectral averaging, and (iii) a generalization of the Dixmier-Fuglede-Kadison differentiation formula.