Yes. Under emperor Justinian, the Byzantines took back much of the Italian peninsula and north Africa. As Justinian was preparing the next assault, a devastating famine struck the Mediterranean ca. 537, which was followed by a severe outbreak of Y. Pestis - the same microbe which caused the Black Death - in 543. Both of these events set the Empire back on its heels, and Justinian died before things recovered enough to make another attempt. For various reasons, his successors had other priorities. The Empire nevertheless persisted to hold territory in Italy, and the Exarchate, the imperial administrative center in Ravenna, persisted until it was finally conquered in 752. The fall of Ravenna is usually considered the final nail in the coffin of Byzantine attempts to reconquer Rome.

Furthermore is there any literature that shows how prominent byzantines felt about losing rome? Did they feel like they lost some sort of culture or importance? Or by that time had there been a big enough difference between east/west empires