NEW YORK (CNNMoney.com) – It has been a week since billionaire activist shareholder Carl Icahn and investment bank Lazard proposed that media giant Time Warner be split into four companies.

Yet so far, several big shareholders of Time Warner (Research) have said they are not impressed with Icahn's plan.

Time Warner CEO Dick Parsons has had to deal with criticism from activist shareholder Carl Icahn. But so far, few investors have said they back Icahn's breakup plan for Time Warner.

Investors bide their Time: Shares of media titan Time Warner have lagged the broader market for the past year.

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"I have to echo the comments of those who would argue this plan is dead on arrival," said Mario Gabelli, chairman and CEO of money management firm GAMCO Investors, which owns about 15.4 million shares of Time Warner, according to data from FactSet Research. "It's got no zip to it. We need to see something else."

Icahn heads a group that controls more than 3 percent of Time Warner's shares and hopes to take over control of the company's board through a proxy fight at its next shareholder meeting later this year. But in order to do so, he's going to need to attract more support from large institutional owners of the stock.

Wendell Birkhofer, a portfolio manager with Dodge & Cox, a San Francisco-based money management firm that owns nearly 137 million shares of Time Warner, according to FactSet Research, said that his firm was "watching the Icahn situation with lots of interest."

He added, though, that his company was not aligned with the Icahn group. He wouldn't comment about specifics of the breakup plan.

Shares of Time Warner fell 10 percent last year due to concerns about sluggish growth prospects in the media business. Many investors have expressed dismay about the fact that Time Warner's stock has been a laggard. And some have voted with their feet.

According to a regulatory filing late last week, Capital Research and Management Co., the company's largest institutional shareholder, reduced its stake in Time Warner by nearly 50 million shares during the fourth quarter of last year. The Los Angeles-based firm still owns about 191 million shares of Time Warner. (Time Warner is the parent of CNNMoney.com.)

But shares have fallen more than 2 percent since Icahn's press conference last week -- a possible sign that Wall Street doesn't think a breakup is the way to boost the stock price.

A hedge fund manager who owns shares of Time Warner said Icahn faces an uphill battle in convincing institutions to back him. "It doesn't seem like there is a whole lot of steam building for his proposal," the manager, who asked not to be named, said.

"People generally believe the stock is cheap and some of the things Icahn said are good," the manager added. "Obviously, he wants to get the stock up but it's not totally clear that a breakup would work."

Time Warner CEO Dick Parsons has maintained that the company is better off with a conglomerate structure rather than separate publicly traded firms for its AOL online unit, publishing division and film and TV operations, as Icahn proposes. Time Warner does plan to spin-off a small portion of its Time Warner Cable unit later this year -- just not the entire division as Icahn urges.

One media analyst at an institutional firm which owns a stake in Time Warner said the biggest problem with the Lazard report -- which weighed in at a hefty 343 pages -- was that there wasn't any big revelation in it about why Time Warner's stock has underperformed the market.

"I'm stunned about how little interesting information is in the report. It is stuff that we already know," said the analyst, who asked not to be named, adding that it was unfair for Icahn and Lazard to blame Parsons for the malaise that's affecting the entire media sector.

Shares of top Time Warner rivals Walt Disney (Research), Viacom (Research) and News Corp. (Research) also underperformed the market in 2005.

The analyst added, however, that pressure from Icahn might be a good thing for shareholders because it could keep Time Warner from making a bold move that might hurt profits, such as making a takeover bid for cable company Cablevision (Research) or Spanish language broadcaster Univision, which recently put itself up for sale.

"Icahn keeps them from doing anything crazy," the analyst said.

Another fund manager with a firm that has a large Time Warner stake agreed that Icahn's presence as a vocal critic of the company could help. The manager said Icahn's comments about how Time Warner could do a better job of keeping expenses in check are valid.

"I don't mind having Icahn around keeping management honest," the manager said. "On the cost side, the company could use some encouragement on showing more discipline."

With that in mind, it's premature to count out Icahn just yet.

Greg Taxin, chief executive officer with Glass, Lewis & Co., an institutional investor advisory firm, said that Icahn has had a recent track record of success with investments in oil and gas company Kerr-McGee and luxury hotel chain Fairmont Hotels & Resorts. So that might sway some institutional shareholders that have yet to make up their mind about who to support in a proxy battle, he said.

"The case that was made by Icahn was certainly a rational, well-considered view and many investors haven't had a chance to digest it," Taxin said. "Icahn has had victories at Kerr-McGee and Fairmont."

Of course, Icahn is far from perfect. Taxin also points out that Icahn's investment in video store rental chain Blockbuster has not panned out.

Chris Young, director and head of merger and acquisitions research with Institutional Shareholder Services, another influential investor advisory group, said he thinks many major Time Warner shareholders are playing their cards close to the vest.

In particular, Young said many institutions won't make up their mind until they find out who Icahn plans to nominate for the board beyond Frank Biondi, the veteran media executive who Icahn has tapped to lead the breakup of Time Warner if he succeeds in ousting the company's current management team.

Icahn promised investors at last week's press conference that he would announce his slate of board candidates either this week or next week.

"The dissidents have their work cut out of them," Young conceded. "But people aren't going to commit until they meet all the actors in this play. It's too early to call."

To that end, although Gabelli said he is currently in favor of Time Warner's management team, led by Parsons and COO Jeffrey Bewkes, he added that he has yet to make up his mind as to how his firm would vote at the shareholder meeting and that the decision would partly depend on who Icahn nominates to the board.

"I like existing management's plan for the company but that doesn't mean we know how we're going to vote yet," Gabelli said.

For more about Icahn's plan to break up Time Warner, click here.

The CEO of cable company Comcast defended Dick Parsons last week. Click here for more.

The reporter of this story owns shares of Time Warner and the Dodge & Cox Stock mutual fund through his company's 401(k) plan.