This is Theodore Metochites, who ordered the restoration of the church and the mosaics and frescoes to go along, kneeling before Christ, whom he offers a model of the church. He is dressed in his official robes, the turban-like hat was called a skiadon (literally a sunshade).
Theodore fell from royal favour in his later years, stripped of his wealth and thrown into prison he was only released when his life was drawing to an end, and he retired to the monastery of this very church, where he died on 13 March 1331. He hoped the church he built would secure him "a glorious memory among posterity to the end of the world." The "Strolling through Istanbul" guide I here quote concludes "It was indeed." I am a bit more of an optimist regarding the prospective life of this world, but do agree to a certain extent.