  We construct Q-operators for the open spin-1/2 XXX Heisenberg spin chain with
diagonal boundary matrices. The Q-operators are defined as traces over an
infinite-dimensional auxiliary space involving novel types of reflection
operators derived from the boundary Yang-Baxter equation. We argue that the
Q-operators defined in this way are polynomials in the spectral parameter and
show that they commute with transfer matrix. Finally, we prove that the
Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their
eigenvalues in terms of the Bethe roots.
