Derivative of a trace function

Question

Let $K$ be a Hermitian matrix, and $X$ be a positive one. What is the derivative of the trace function $$ \mbox{ Tr } X|e^{itK} - X|^3$$ with respect to $t$ at $t = 0$ ? There is a nice formula for this?

Answer 1

Let $A=e^{itK}-X$; then $A^*=e^{-itK}-X,A'=iKe^{itK},A'(0)=ik$. Let $A^*A=V,U=V^{1/2}$. We consider $f(t)=tr(XU^3)$; then $f'(0)=tr(X(U'U²+UU'U+U²U'))_{|t=0}$. It remains to calculate $U'(0)$. Note that $V(0)=(I-X)^2$ is $>0$ iff $(*)$ $1\notin spectrum(X)$.

Under the condition $(*)$, $U'(0)=\int_0^{\infty}e^{-xV^{1/2}}V'(0)e^{-xV^{1/2}}dx$. Clearly $V'(0)=(A'^*A+A^*A')_{|t=0}=-iK(I-X)+(I-X)iK=0$; finally $U'(0)=0$ and $f'(0)=0$.