Positivity of a quadratic polynomial across a spectral gap ($(B-aI)(B-bI)>0\text{if }\mathrm{spec}B\cap [a,b]=\varnothing$)

Let $B$ be a finite-dimensional Hermitian matrix and $a<b$ real. The spectral theorem gives $v={\sum }_j{c}_j{x}_j$ in an orthonormal eigenbasis, soIf the spectrum avoids the closed interval, every factor product is positive. Consequently the quadratic form is positive for $v\ne 0$ and zero for $v=0$. More precisely it is bounded below by ${\min }_j[({\lambda }_j-a)({\lambda }_j-b)]\Vert v{\Vert }^2$. This is a finite-dimensional instance of testing a spectral gap with a polynomial in the operator.