Anyone have any primary source Muslim writing on the Byzantines? I'd love to learn more.
I read several ones, including several excerpts from Tabari and Qazvini among others, and I have some in the other house -the one i'm not currently living in.
If my memory helps, the Muslim attitude was a mix of rivalry, admiration, and respect. In the Western sources, OTOH, my impression is that contempt prevailed over admiration: the "oriental luxury" seemed a signal of decadence to many observers.
It is important to note that under Umayyad rule, i.e. exactly when the wars between the Caliphate and the ERE were more common and more desperate (Byzantines were actually fighting for survival, and so probably felt the Muslims too) there was also a great amount of cultural exchange. The Arabs felt that the Byzantines had something to teach them and were on the average eager to learn, but also, to adapt what the Byzantines had to specically Muslim needs. It was clear especially in two fields: art and law.
Later on, under the Abbasids, the Muslim empire grew in self-confidence and saw the Byzantines as the unworthy inheritors of the ancient greek greatness; but still, the city itself and, IIRC, the Orthodox liturgy were seen as marvels. Under the Abbasids there was a great deal of continuity and reprise of cultural attitudes of the Sassanian court, and the re-appropriation of the Greek heritage played a great role in it. The Muslim elites saw themselves as enlightetened patrons of knowledge as opposed to the decaded, ignorant bigots the Greeks had become under that pervert system of belief their treacherous priests had made out of Christianity. Though Christianity in itself was respected, its Byzantine form was seen as a paganized bigotry from an Abbasid court perspective.
However, the Abbasids still kept the old persian ideal of the "twin civilized empires" as it was explicitly described by Simocatta (that put the expression on Khusraw Parviz's mouth). The Khazars and the Aksumites were sometimes included into the system, but for all intents and purposes, Byzantium was the Great Civilized Other in the eyes of both Persians and Muslims.
Later sources sometimes give a more exhalted representantion of Christianity, especially in context of rebuttal of dualist heresy (Zandaqa), Mazdeism, or Gnosticism, as the Christians were seen at least potentially as fellow monotheists (a fact that, on the other side, was recognized by Pope Gregory VII) though deviated by ignorant monks.
Putting the Jewish Khazars into the equation,as authors such as Ibn Fadlan and Ibn Rusta did (i've read excerpts from both) generally tends to improve the view on Byzantium by comparison. Sometimes the Khazars were seen as civilized as well, sometimes not.
On average, I'd say Byzantium was depicted as richer in goods, and Baghdad (or whatever) as richer in knowledge and skills (including tech: high tech automata were built to impress those ignorant Christians, both Eastern and Western).
For the apocalyptic discourse, I still don't know enough.