Scholars assign most of Isaiah 1-39 to the prophet in the 8th century BC. Chapters 40-55 (Deutero-Isaiah) are considered to be by a later hand in the 6th century, and chapters 56-66 (Trito-Isaiah) some time later still. The main arguments are out of language and style, but ultimately, they and others rest on very shaky assumptions.
Language
The poetic nature of Deutero-Isaiah makes dating it more difficult. [Arentsen, 511, 514, 525] As a result, a lot of the conclusions from its language are probably inconclusive.
Participles
Some see the high amount of participles to imply a later author: pre-exilic writings prefer the finite verb. But in prophecy, which is in poetry, the number of participles rise. Deutero-Isaiah may still have a higher percentage of participles than books like Hosea or the pre-exilic Psalms, but the highest concentration of participles is in the indisputably pre-exilic book of Nahum, so this argument can't be used with any certainty. Job, often considered linguistically early, uses more participles than any late biblical Hebrew book. In Deut-Isaiah, there are frequent poetic participles describing God. Used for timeless truths and ongoing divine action. Not used heavily for prose predicate replacement (unlike LBH prose). Jeremiah uses moderately more participles than his prose, for example.
Some older scholarship tried associating high participle count with lateness, but newer studies (Hurvitz, Joosten, Holmstedt, Muraoka, Young–Rezetko–Ehrensvärd) agree it's a stylistic/genre feature. This criteria isn't even on a typical list of late biblical Hebrew features. [Kofoed, 144]
Syntax
Isaiah 40-55 retains classical prophetic syntax, but with looser parallelism: looser bicola (sometimes tricola), extended particles, and some verbless clauses functioning like later elliptical constructions as well as occasional participial predicates.
But this is all due to the prophetic genre and thus poetic style of the book, because Ezekiel, a later book, has more rigid syntax, far fewer extended particles, and more participial predicates than Deutero-Isaiah. Jeremiah has fewer asyndetic imperfect apodoses than Isaiah 1-39 where they appear regularly - mainly due to the oracles and poetry. In fact, Deut-Isaiah and First Isaiah have the same percent of elliptic verbless clauses in prose. This should point to the same author as Deutero-Isaiah. Participial predicates, whereas we see some of this in Nahum, a pre-exilic book. Deutero-Isaiah has the same percent of elliptic verbless clauses in prose as Isaiah 1-39.
Grammar
The waw-consecutive is infrequent in Deutero-Isaiah because of the highly poetic style. The fact that wayyiqtol is sometimes used, making the frequency about the same as in Jeremiah which is a mixture of prose and poetry (and one expects wayyiqtol more in prose) should mean Deutero-Isaiah shouldn't be dated later than Jeremiah on this basis.
The use of תℵ and םע is ambiguous. Arentsen uses the higher frequency of םע to argue that Deutero-Isaiah stands somewhere between Jeremiah and Ezekiel. But this would make Hosea late biblical Hebrew, and Ruth, which evenly uses both in his chart, pre-exilic! [Arentsen, 515] The fact is, many of the so-called late biblical features are difficult to apply to poetry. [Arentsen, 513-4] This is also true of various placements of the object, [Arentsen, 513] or the classical Verb-Subject-Object (VSO) and the later Subject-Verb-Object (SVO): SVO is used more frequently in poetry.
Vocabulary
The plene (later, formal spelling) may be high for Deutero-Isaiah, but First Isaiah has it lower than the earlier books of Hosea and Amos. Popular books or sections would've been copied more frequently, resulting in those sections being rewritten in the language of the time, as we see in the Minor Isaiah Scroll. Deutero-Isaiah uses the older spelling of David the one place he's mentioned. This isn't a swaying argument because Ezekiel uses the older spelling three out of four times, but it does mean the plene argument shouldn't be used too seriously by itself. Ezra and 2 Chronicles, much later in their language in other regards, have the same percentage of plene as Ezekiel and Deut-Isaiah. Dt-Isaiah uses slightly more waw and yod to mark vowels than First Isaiah, but nowhere near Ezra-Nehemiah for example. Since the difference between the exilic Ezekiel and post-Exilic Ezra/2 Chron isn't much bigger, the plene percentage of Dt-Isaiah being the same as later books is very possibly due to later copying.
Aramaisms
Aramaisms, especially in poetry, do not prove a late date. [Arentsen, 522] The highly poetic nature of Deutero-Isaiah could explain the larger number of Aramaisms. These are about as many as Jeremiah, yet Deut.-Isaiah is much more poetic, suggesting it's an earlier work.
Older Language in Isaiah 40-66
The language of all of Isaiah 40-66 is pre-exilic, as Mark F. Rooker has shown. [Rooker, 303-12] There are also stylistic similarities (e.g. "double" punishment in Isaiah 40:2 and 61:7) that suggest the same author for Deutero and Trito Isaiah.
Other Objections
Will Zion fall?
One claim says the actual prophet Isaiah did not believe the fall of Jerusalem was possible, nor any prophet until Jeremiah. By this logic, neither does Deutero-Isaiah (e.g. Is. 49:16). At any rate, the claim is untrue (Isaiah 22:9), and the punishments in the Torah are very explicitly outlined. If the Assyrians could destroy the Northern kingdom, no one could doubt it could happen to the South.
It's objected that Jeremiah didn't cite earlier prophets on the possibility of the destruction of Jerusalem (e.g. in Dt-Isaiah). But Jeremiah's opponents don't mention the supposed belief in an invincible Jerusalem.
Copying Jeremiah
It's a struggle to see any dependence on Jeremiah aside from common prophetic words. Just about the only arguable case is from Jeremiah 16:18/17:18 and Isaiah 40:2/61:7 where the Israelites are to be punished "double" for their sins. But Dt-Isaiah (and Jeremiah) could have easily taken this from the Torah (Ex. 22:4, 7, 9; cf. Zech. 9:12). It's not like Dt-Isaiah was trying to hide any of his sources if he had any.
Genuinely (near)-verbatim verses shared between Old Testament prophets exist whereas the Jeremiah-Dt Isaiah example doesn't even come close. For example, Isaiah 2:2-4 is virtually word for word with Micah 4:1-3. Jeremiah 49:14-16 quite possibly borrows from Obadiah 1:1-4. Jeremiah 26:18 directly cites Micah 3:12.
And then there are common prophetic traditional phrases and terms: Isaiah 13:6 and Joel 1:15, Ezekiel 38-9 with Joel 2-3, Amos 5:18-20 and Isaiah 5:30, Hosea 1:10, 2:1 and Isaiah 10:22-3. Similar such poetic tropes are detectable in the Iliad for example - Homer did not come up with everything but used the common stock tradition. These are common ways to express that we use today as well ("in any case", "it so happened", "the pot calling the kettle black" etc). These are not copied prophecies, these are ways of expressing one's thought that's familiar that doesn't reinvent the wheel, or there'd be much more literary parallels beyond a phrase or two.
Past tenses
The argument that most of the prophecies in Isaiah 40-66 are written in the past tense as if they had already occurred is untrue. The dominant tense in both First and Deutero Isaiah is the imperfect (used for the future), not the perfect/completed (past). It is true that Dt-Isaiah has twice as much prophecies in the past tense by percentage, but that is because of poetic judgment or salvation just like in First Isaiah (e.g. Isaiah 9, 10, 21).
Prophecies in the past tense reflect something that the author has seen. Thus, for example, Revelation's prophecies are very much in the past tense. In addition, poetry leans toward variability: "a poet may for the sake of variety alternate the tense representing an action as past and another closely connected to it as present." [Saydon, 291]
Evidence of Inspiration
Isaiah 13-14, 21 predict the fall of Babylon. These chapters are written in firmly Middle Classical Hebrew, and are earlier than Nahum (late 7th century BC) and Jeremiah. Scholars presume that the word "Babylon" was interpolated, which goes to show their a priori assumptions. The manuscript tradition does not reflect this, nor does it make sense for three separate prophecies to have it interpolated.
Nor is the prophecy an educated guess. The fall is described as being "near" (Is. 13:6, 22). Empires lasted for centuries. The Neo-Assyrian Empire lasted 250 years, the Medes were also there for hundreds of years. For an author to predict an impending fall under Nebuchadnezzar would be like supposing America completely disintegrated a few generations after WWII (e.g. the year 2000). Quite simply, it would be some luck to predict Babylon's fall (three times no less!) and get it right.
Suggestions that this was about the independent Babylon of 703-2 under Marduk-apla-iddina II who visited Hezekiah is very doubtful since it was a very brief independence, not leading "nations" (Isaiah 13:4).