“It’s about being able to tap into trends — not just fashion trends, but what people want in their lives, what they’re talking about — while maintaining a sense of the brand,” said Simon Ungless, director of fashion at the Academy of Art University in the San Francisco area, who has worked for designers like Alexander McQueen. “I think that’s what she’s done at Burberry very well, and that feels like something they’ve lost at Apple.”

Ms. Ahrendts, 53, was raised far from the runways. She grew up in Indiana and graduated from Ball State University, also in Indiana. She eventually found her way to the fashion industry and started rising up the executive ranks, becoming Burberry’s chief executive in 2006.

As Burberry’s fortunes rose, so did Ms. Ahrendts’s profile. That profile could grow even larger at Apple, one of the biggest companies in the world — and one of the most closely followed. She will now be the only woman on Apple’s executive team, and some analysts on Tuesday speculated that she could be an eventual successor to Mr. Cook, though he is not expected to leave anytime soon.

Ms. Ahrendts, who joined Burberry from Liz Claiborne, is Apple’s second big hire from the fashion industry this year. In the summer, Apple hired Paul Deneve, the former chief executive of the French fashion house Yves Saint Laurent, to work on special projects.

Silicon Valley has long had its eye on fashion and the industry’s success at selling an image. And the push by technology companies into so-called wearable technology, like Google Glass and Samsung’s smartwatch, has only intensified that interest. When models wearing Google’s eyewear walked the runway at Diane von Furstenberg’s show in New York this year, it became obvious that the connection was lasting.

The hiring of Ms. Ahrendts increased expectations among analysts that Apple would expand into wearable technology, including perhaps an Internet-connected wristwatch.