  We construct a Q-operator for the open XXZ Heisenberg quantum spin chain with
diagonal boundary conditions and give a rigorous derivation of Baxter's TQ
relation. Key roles in the theory are played by a particular
infinite-dimensional solution of the reflection equation and by short exact
sequences of intertwiners of the standard Borel subalgebras of
$U_q(\widehat{\mathfrak{sl}_2})$. The resulting Bethe equations are the same as
those arising from Sklyanin's algebraic Bethe ansatz.
