UPDATE 2-Google CEO discusses Apple board ties, Chrome

* Says economy's worst over but no strong sign of recovery

* Chrome on Microsoft turf but Google downplays rivalry

* Schmidt will talk to Apple about recusing from board

* Schmidt was kept informed about Jobs' medical leave

(Adds comments about Chrome, additional details)

By Alexei Oreskovic and Yinka Adegoke

SUN VALLEY, Idaho, July 9 (Reuters) - Google Inc (GOOG.O) Chief Executive Eric Schmidt said he will talk to Apple Inc (AAPL.O) 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 (MSFT.O) 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.