SUN VALLEY, Idaho (Reuters) - Google Inc Chief Executive Eric Schmidt said he will talk to Apple Inc about how his role on its board might change after Google’s move to launch a new operating system.

Earlier this week, Google announced it was developing software for personal computers based on its Chrome Web browser, which would compete with Microsoft Corp’s Windows as well as Apple’s OS X platform.

Schmidt said at the Allen & Co media and technology conference in Sun Valley, Idaho, on Thursday that PC makers could announce products running Chrome OS later this year.

Schmidt, whose company has weathered the recession better than most technology and media companies, also said that while the worst of the economic downturn has passed, there aren’t any strong signs of an immediate recovery.

There had been talk during one presentation at Sun Valley of a “square root recovery,” meaning the economy could rebound but then remain flat, like the shape of the square root sign, Schmidt said.

“We’re through the shock, we’re through the collapse. I would argue the worst is behind us, speaking generally, not specifically to Google,” he said at a one-hour briefing with reporters, along with Google co-founder Larry Page.

But much of the focus of the briefing was on Google’s new operating system. Because it would compete with Apple’s own software, Schmidt said he would talk to the Mac computer maker about whether he should recuse himself from Apple’s board.

Under federal antitrust law, a person is not allowed to sit on the board of two companies if it decreases competition between them.

“I’ll talk to the Apple people. At the moment, there’s no change,” Schmidt said.

Google, the No. 1 Internet search company in the U.S., is under increasing regulatory scrutiny. The U.S. Federal Trade Commission is looking into whether ties between the boards of Apple and Google violate antitrust laws. Schmidt and former Genentech CEO Arthur Levinson are directors of both companies.

Schmidt began recusing himself from Apple board meetings where the iPhone was discussed after Google launched its own Android mobile phone operating system.

INCREASING COMPETITION

Google has said that computer makers like Hewlett-Packard Co and Acer Inc are working with it on Chrome devices, with the first netbook PCs expected in the second half of 2010.

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“Everybody we’ve talked to under nondisclosure is excited about the plan. So hopefully later this year we’ll see some announcements,” Schmidt said.

The move intensifies the competition between Google and Microsoft, whose Windows operating system runs on the majority of the world’s personal computers. And it comes in the wake of Microsoft’s recent push into Google’s home turf with the release of the Bing Internet search engine.

Schmidt and Page repeatedly steered the conversation away from the rivalry with Microsoft, describing the Chrome OS as a product that would be “additive” to the PC market.

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“I don’t want to talk about Microsoft,” Schmidt said, noting later that Google did not have any particular goals for Chrome to take market share in the PC business. “We actually don’t look at market share at all,” he said.

The Chrome OS is a separate product than Android, the operating system that Google released last year aimed primarily at smartphones. But Schmidt said the two products are closely related and could eventually “merge even closer.”

Schmidt also touched upon other business efforts, and cited display advertising as the primary revenue opportunity after Google’s $3.2 billion acquisition of DoubleClick in 2008.

“I would suggest the display part of our business is likely to be the next billion dollar business,” he said without giving a timeframe.

Schmidt also said he was kept well abreast of the circumstances surrounding Apple founder and CEO Steve Jobs’ medical leave of absence, which began in January. Jobs received a liver transplant while on leave, and has returned to work.

Apple and its board have been criticized by some observers for failing to disclose the exact state of Jobs’ health.

“I was extremely well-informed as a board member with what was going on with Steve,” Schmidt said. He declined to comment further on the matter.