Arteta was confirmed as the new Arsenal manager on Friday after he agreed a three-and-a-half year deal earlier in the week.
Kroenke said: "We're delighted to be bringing Mikel back to Arsenal.
"He knows our expectations and those of our fans around the world are high and we are confident he can play a lead role in taking the club back to the levels we all demand.
"I also want to thank Freddie Ljungberg for skilfully guiding us through the last three weeks.
"He stepped up at short notice and has helped us through this difficult period with great professionalism."