The 28-year-old Dutch forward misses the first leg of the play-off against Serie A side Udinese at Emirates Stadium later on Tuesday as he is serving a one-match ban after being sent off against Barcelona in last season's Champions League round-of-16 tie.
He will have to wait until Saturday, when Arsenal play Liverpool at Emirates Stadium, for the armband passed to him following Fabregas's departure to Barcelona on Monday.
Van Persie, who has been at Arsenal for seven years, told the club's website: "As captain you are an ambassador for the Club and I'm ready for that.
"I've been in the game now for more than 10 years, so I know how this world works. When I was 18 I didn't have a clue... but now I feel experienced, so I feel ready for this."
As well as Van Persie sitting out the match, manager Arsene Wenger is serving a one-match touchline ban for comments he made to referee Massimo Busacca following Van Persie's dismissal against Barcelona last March.
Samir Nasri, who could be leaving the club this week for Manchester City and who criticised Arsenal's fans for being "disrespectful" to him during Saturday's match at Newcastle, is also suspended and misses the game.
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