Murray, a three-time grand slam champion, is not targeting a specific return to action, but has earmarked the start of the grass court season as a realistic aim.
Murray, 30, who has not played competitively since last year's Wimbledon, made the announcement on his Facebook page hours after the surgery and says he is confident he will return as strong as before.
Murray has previously suggested that he would only consider an operation on the area as a last resort, but the Scot has now revealed that he is optimistic that he can make a return to professional tennis, potentially during the grass-court campaign. Thanks to everyone for the well wishes and support the last few days.
"I want to know when I come back that I'm ready", he said.
Murray suffered a major back injury in 2013 and underwent surgery following the US Open, missing the end of the season, to return in January 2014.
He went on to win a second Wimbledon title in 2016 and played arguably the best tennis of his career.
He has not played competitively since Wimbledon past year after first feeling pain in his hips at the French Open.
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In an emotional post on social media last week, Murray stressed he wanted to avoid undergoing surgery but it was seen as the only option after six months of painstaking rehabilitation work hadn't repaired the damage.
"I want to come back when I'm fit and ready to play, not to get into a situation like in Brisbane or NY, where I'm unsure when I turn up at a tournament how fit I am". "Obviously, I was still doing fine a year ago - I was ranked No 1 in the world".
Murray also said he underwent some minor surgery on his groin on 18 December.
"So if I can get myself to 95% of my best, I believe that's enough to compete at the highest level".
Murray said: "I've been fairly competitive with top-50 players in the world practising in Brisbane when I was struggling to move".
Murray is now back in the United Kingdom and before leaving Australia stated that he would be considering all the options to try and find a solution to an issue that has kept him out of tennis since the Wimbledon quarter final in July.
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