We know this mosaic was completed in the 11th century because it depicts the Empress Zoe and her third husband, Constantine IX who reigned between 1042-55 CE:
The mosaic was actually completed earlier during the lifetime of Zoe's first husband. It underwent repeated head upgrades when she remarried (in dubious circumstances).
Finally, this is a beautiful 13th century mosaic of the "Deisis"--Christ at the judgement with Mary petitioning for the world's salvation on his right, and John the Baptist on his left:
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