  The paper deals with multidimensional Bochner-Phillips functional calculus.
In the previous paper by the author bounded perturbations of Bernstein
functions of several commuting semigroup generators on Banach spaces where
considered, conditions for Lipschitzness and estimates for the norm of
commutators of such functions where proved. Also in the one-dimensional case
the Frechet differentiability of Bernstein functions of semigroup generators on
Banach spaces where proved and a generalization of Livschits-Kre\u{i}n trace
formula derived. The aim of the present paper is to prove the Frechet
differentiability of operator Bernstein functions and the Livschits-Kre\u{i}n
trace formula in the multidimensional setting.
