The 24-year-old Arvidsson, tied for the Predators lead in goals (31) and points (61) during the 2016-17 season and added 13 points in 22 playoff games.
Martinook scored 11 goals and added 14 in 77 with the Coyotes last season.
Calgary selected the American-born keeper in the third round of the 2012 draft. Gillies has played one game at the NHL level so far.
Rittich, an undrafted Czech, also made one appearance in the NHL last season and posted a 15-11-6 record for the AHL's Stockton Heat.