The Fock operator, which appears in the widely used Hartree-Fock theory and Kohn-Sham density functional theory with hybrid exchange-correlation functionals, plays a central role modern quantum chemistry and materials science. The computational cost associated with the Fock exchange operator is however very high. In a simplified setting, the Hartree-Fock equation requires the computation of low-lying eigenpairs of a large matrix in the form A+B. Here applying A to a vector is easy but A has a large spectral radius, while applying B (the Fock operator) is costly but B has a small spectral radius. It turns out that most eigensolvers are not well equipped to solve such problems efficiently. We develop an adaptive compressed method to efficiently treat such eigenvalue problems. We prove that the method converges locally, and surprisingly, converges globally from almost everywhere. The adaptive compression method has been adopted in community electronic structure software packages such as Quantum ESPRESSO, and offers an order of magnitude speedup compared to existing methods. We also demonstrate that the adaptive compression method can enable hybrid functional calculations in planewave basis sets with more than 4000 atoms, and discuss the possible applications of the adaptive compression method beyond the Hartree-Fock-like equations.