Extrapolation for bilinear compact operators in the variable exponent setting

We establish extrapolation of compactness for bilinear operators in the scale of weighted variable exponent Lebesgue spaces. First, we prove an abstract principle relying on the Cobos-Fern\'{a}ndez-Cabrera-Mart\'{i}nez theorem. Then, as an application we deduce new compactness results for the commutators of bilinear $\omega$-Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers acting on weighted variable exponent Lebesgue spaces. Our work extends and unifies among others earlier works of the second named author together with Hyt\"{o}nen as well as Oikari.