Every once in a while I'll come across some feature on the history or exploits of Constantinople. They never fail to mention that Byzantium was the Eastern half/division/branch of Rome.
But I've never gotten a decent explanation of what that means, and Wikipedia doesn't help much. Does that just mean that the Byzantine territories were Roman provinces (in the way that you might call North Africa "the Southern half of the Roman Empire)? Or was there some other dynamic involved?
If it was territory controlled by the same Romans of Italy you would think that after the fall of the city the Empire would have just shifted its administrative infrastructure to its other metropolis and continued. As far as I know that wasn't the case however.
In the late 3rd century, after a long series of crises, the Roman emperor Diocletian instituted a number of reforms to try to strengthen and stabilize the empire. Among them was the idea of having four emperors—two junior and two senior—who would be better able to administer and defend all the empire’s territories.
This didn’t work out quite like Diocletian had intended; but over the next century it became more and more common for the office to be shared between multiple emperors. Another key step came when Constantine, after converting to Christianity, designated Byzantium as a second capital with an equal status to Rome, with its own Senate and imperial court but with Christianity replacing Rome’s older religious traditions.
At the end of the 4th century the last sole emperor, Theodosius, died, dividing the empire between his two sons. The “Byzantine” empire is the term eventually applied to the eastern division; but there was no difference in status between the eastern and western divisions. The eastern division continued to call itself “Roman”, and was recognized as a division of the Roman Empire by the western empire as long as it endured.
The eastern division lasted a long time, though; and it continued to evolve (as the early empire had always been evolving even before the split). By the end of its existence, it was different enough from the early Roman Empire that historians found it convenient to use a different term to clarify which era of its history they were referring to.
If it was territory controlled by the same Romans of Italy you would think that after the fall of the city the Empire would have just shifted its administrative infrastructure to its other metropolis and continued. As far as I know that wasn't the case however.
There are some misconceptions here. By 4th century AD, the Empire was not controlled by "Romans of Italy". All free inhabitants of the Empire were Roman citizens whether they lived in the city of Rome, or former "provinces". In fact the city of Rome wasn't even the seat of the Emperor for most of the time after 200 AD. Constantinople - the New Rome became the most significant city of the Empire in mid 4th century and remained its capital until 1453 when it fell to the Ottomans.
Before Diocletian in the late 3rd Century, the Roman Empire was essentially unified and administered, at least in theory from the city of Rome. Diocletian reorganized how the empire was run in a number of ways including dividing it into Western and Eastern parts to be administered by separate emperors, and by the time of Constantine, the Eastern part was ruled from Constantinople. This lasted for about another 175 years or so, with essentially two separate but related empires, until the Western part dissolved into a number of small kingdoms which became the ancestors of the modern Western European nations. The Eastern part continued for another thousand years, calling itself Roman in spite of the fact that they rarely had control of the city of Rome.