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

A Q-operator for open spin chains I: Baxter's TQ relation