Both Uses of

acknowledge
in
The Two Towers

I will forget my wrath for a while, Eomer son of Eomund,' said Gimli; 'but if ever you chance to see the Lady Galadriel with your eyes, then you shall acknowledge her the fairest of ladies, or our friendship will end.†
Chpt 3.6 *

acknowledge = recognize

In those days our bounds were away south beyond the mouths of Anduin, and Umbar, the nearest of their realms, acknowledged our sway.†
Chpt 4.4

acknowledged = recognized

Definitions:

(1) (acknowledge as in: acknowledge her or the truth)
express recognition or appreciation of someone or something; or admit something
(2) (meaning too rare to warrant focus)
meaning too rare to warrant focus:

Specialized senses include sending an "acknowledgement" (letter or other message) to indicate that something was received and the "acknowledgements" section at the beginning of a book where an author recognizes and thanks other people who helped in creating the book or ideas contained in it.