Yes. From around the end of the Roman Republic to the opening up of Atlantic Trade, Italy was the most prosperous place in the Mediterranean. It would continue to be fairly prosperous until the Industrial revolution, to which it would instead adapt slowly and inefficiently.

How do we know this? Beyond anecdotal descriptions of expenditure of by surplus income on things like art, and architecture, and other signifiers of material wealth, Economic Historians have spent significant time trying to evaluate historic per-capita economic output. It’s not an easy tasks by any means, and the necessity to maintain academic rigor means that estimates must forgoe decades or even centuries where the quantity or quality of information just isn’t there. Net of this, the estimate stands head and shoulders above the rest cited all over Economic History research was started by Angus Maddison at the University of Groningen. “The Maddison Project” now involves dozens of scholars across the world working to revise and develop its estimations of world economic output across historical eras.

What has the Maddison project concluded? Well, to answer your main question right away, Italy was the most productive economy (per-capita) in Europe up to the mediterranean economy being surpassed by Atlantic trade in the 18th century (and starts falling increasingly behind in lead-up to the Industrial Revolution, even though it still manages to remain remained the third most-prosperous economy in Europe up to around 1820, behind Great Britain and the Low Countries which were instead the easiest industrializers).

Is this all that surprising? The Roman Empire was fundamentally a network of cities, with an institutional setup which could only have emerged in an urbanized social environment (in other words, the “Roman Way of Life,” as imagined by the Empire’s leaders was fundamentally urban life, even in an environment where the main economic activity was agriculture). Italy is also agriculturally fertile and diverse, allowing it to support numerous large cities. Its location in the middle of the Mediterranean also allowed Italian cities to source foodstuffs from other parts of the Mediterranean, allowing its cities to grown even more: At one point, the Roman grain dole was principally sourced from Egypt. Even if the fall of the empire did cause cities to shrink over time, Italian cities would continue to be able to source foodstuffs from across the sea (Tuscany, a comparatively less fertile region, was largely supplied with grain from Sicily, for example).

Cities allow people to specialize in high-value added economic activity, and close proximity fosters inclusiveness in institutions, favoring predictable outcomes, favoring stability, and with that stability, encouraging commerce. Overall, places with dense population are more prosperous than places that are less densely populated: if you look at the most prosperous places during the Roman Empire, interesting patterns emerge: Egypt and the Levant, for example, are places that start the Common Era with high per-capita economic output, only a few levels below Italy's. This is due to agricultural productivity which could support large cities (e.g. cities like Cairo and Alexandria, all the way to Antioch).

One thing which might be causing confusion is the assumption that the Medieval Period represented economic (and social) stagnation. Even if this were true, wouldn’t all of post-Roman Europe regressing equally still mean Italy would be comparatively the most prosperous region? And while it’s not true that technological progress is never linear, and the fall of the empire did usher in an era where war and instability in the old core was more frequent, society also didn’t stop altogether: over the medieval period, peripheral regions of the Mediterranean developed slowly but unmistakably, adopting new institutional innovations and new technologies. The core of the old Empire kept developing as well as well: Italian institutions became increasingly city-centric, fostering trade and competitiveness, and sustaining economic expansion which probably would have progressed into the 16th century had the Italian Wars not broken out (and keep in mind that Italian income was so high, even without growing much it remained one of the the top three per-capita European economies until 1820).

You might be interested in this older answer of mine, which isn't exactly aligned with what you ask but analyzes many of the same principles.