Christopher B. Hays explains multiple authorship of Isaiah.
Christopher B. Hays, "Isaiah," The Oxford Encyclopedia of the Books of the Bible (Oxford: Oxford University Press, 2011)
The book of Isaiah is one of the longest and most diverse in the Bible, comprising sixty-six chapters composed and compiled over the course of multiple centuries. It has thus become a crossroads for historical and literary interpretations, and indeed for practically every method of critical study. The book's formation can even be considered as a kind of microcosm of the formation of the canon as a whole (Blenkinsopp 2002). Furthermore, it has taken on large theological significance in later religious traditions; Isaiah was second only to Moses as a prophet in at least one rabbinic tradition, and the book was esteemed as a “Fifth Gospel” by classical Christian interpreters (see further below, under “History of Interpretation”).
The book's superscription attributes it to “Isaiah son of Amoz.… in the days of Uzziah, Jotham, Ahaz, and Hezekiah, kings of Judah” (1:1) indicating a prophetic career spanning roughly the second half of the eighth century b.c.e. However, even in the premodern period commentators realized that not all of the book was attributable to a single prophet working in that time. In short, while chapters 1–39 contain clear references to events in the Neo-Assyrian period, such as the Syro-Ephraimite war in the 730s and Sennacherib's siege of Jerusalem in 701, chapters 40–48 contain equally clear references to events of the end of the Babylonian period and the beginning of the Persian period, particularly Persia's conquest of Babylon in 539 and the edict of its king, Cyrus, to return the Judeans to their homeland in 538.
The gap of nearly two centuries between the career of Isaiah ben Amoz and the context of 40–66 has been understood—by interpreters both ancient (Josephus, Ant. 11) and modern (Oswalt)—to reflect Isaiah ben Amoz's supernatural foreknowledge. However, the idea that it instead reflects the book's long history of composition and compilation is one of the most widely accepted conclusions of critical biblical scholarship.
A further level of complexity is introduced by the fact that at least chapters 1–39 are widely recognized to have been augmented and edited by later tradents (that is, those who passed on the texts). Because of the intricacies of that topic, the two major sections of the book will be addressed first, followed by a reflection on the implications for theories of the book's formation. Since the influential work of Bernard Duhm more than a century ago, chapters 1–39 have been known as “First Isaiah,” chapters 40–55 as “Deutero-Isaiah” (or “Second Isaiah”), and chapters 56–66 as “Trito-Isaiah” (or “Third Isaiah”).