A proof of the Riemann hypothesis

In this paper we study traces of an integral operator on two orthogonal subspaces of a $L^2$ space. One of the two traces is shown to be zero. Also, we prove that the trace of the operator on the second subspace is nonnegative. Hence, the operator has a nonnegative trace on the $L^2$ space. This implies the positivity of Li's criterion. By Li's criterion, all nontrivial zeros of the Riemann zeta-function lie on the critical line.