Okay, lets assume Basil II lives long enough to take Sicily in 1025. Or maybe (but less likely) George Maniakes succeeds in taking it in 1038. Either way, by the mid-eleventh century, the Catapanate of Italy and Sicily are under Byzantine control. The question is, how long could they keep it.
Keep in mind, while it may be under their control, it probably isn't firmly under Imperial control, and you will have to deal with restless Italians, Lombards, Normans, and possibly Arabs. There are also the Germans/Italians to the north, and North Africans to the south. And of course, all of the problems back home. Still, could they keep it? And if so, what would be the effects?
Keep in mind, while it may be under their control, it probably isn't firmly under Imperial control, and you will have to deal with restless Italians, Lombards, Normans, and possibly Arabs. There are also the Germans/Italians to the north, and North Africans to the south. And of course, all of the problems back home. Still, could they keep it? And if so, what would be the effects?