In the proof of Theorem 3.1 of the paper Inequalities for M-matrices, Ando evaluates a matrix function (see equation boxed in orange below) via an integral representation of a Pick function (see Lemma 3.1):
A well-known faulty argument for the Cayley–Hamilton theorem is to replace $t$ with $B$ in the characteristic polynomial $p(t) := \det (tI - B)$. And, obviously, one can not replace $t$ with $A$ to evaluate $p[A]$. Hopefully, this motivates the following:
Question: What is the justification for being able to evaluate $f[A]$ in this manner?
Ando provides the definition of a matrix function via the Cauchy integral formula on p. 300, but this doesn't seem to be of much help.
