Amos 9:12

12 As a result they
They probably refers to the Israelites or to the Davidic rulers of the future.
will conquer those left in Edom
Heb “take possession of the remnant of Edom”; NASB, NIV, NRSV “possess the remnant of Edom.”

and all the nations subject to my rule.”
Heb “nations over whom my name is proclaimed.” The Hebrew idiom indicates ownership, sometimes as a result of conquest. See 2 Sam 12:28.
This verse envisions a new era of Israelite rule, perhaps patterned after David’s imperialistic successes (see 2 Sam 8–10). At the same time, however, the verse does not specify how this rule is to be accomplished. Note that the book ends with a description of peace and abundance, and its final reference to God (v. 15) does not include the epithet “the Lord who commands armies,” which has militaristic overtones. This is quite a different scene than what the book began with: nations at war and standing under the judgment of God.

The Lord, who is about to do this, is speaking!
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