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

TL;DR

This paper extends the theory of perturbation determinants and operator differentiability within the Hirsch functional calculus on Banach spaces, focusing on nonpositive operators with nuclear differences.

Contribution

It generalizes the logarithmic derivative formula for perturbation determinants and investigates Frechet differentiability of operator monotonic functions on Banach spaces.

Findings

Derived a generalized formula for the logarithmic derivative of perturbation determinants.

Proved Frechet differentiability of operator monotonic functions on Banach spaces.

Contributed to the development of Hirsch functional calculus.

Abstract

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.