Consider the following setting:
Let $M$ be a smooth manifold and suppose that $T \colon M \to T^{(1,1)}(M)$ is a smooth section, that is, $T_x \in \mathrm{End}(T_xM)$ for every $x \in M$. Moreover, suppose that $T$ is fibrewise symmetric and fibrewise positive. Hence we have that the spectrum $\sigma(T_x)$ is contained in $(0,+\infty)$ for all $x \in M$.
Now, for every $x_0 \in M$ there should be a neighbourhood of $x_0$ in $M$ such that the spectrum $\sigma(T_x)$ is contained in $[a_{x_0},b_{x_0}]$ with $a_{x_0} > 0$ for every $x$ in this neighbourhood. My reasoning is that this should follow from the continuous dependence of the roots of a polynomial of the coefficients. Is that true? Are there any good references on this (a book)? Because I've mostly found papaers on this subject.