Turkey is important US NATO ally. To imagine that the US isn't closely involved in this Turkish "invasion" seems highly unlikely.
The Turks have a problem- and it's not a new one. It's in US interests to do what it can to resolve this problem.
Whether the military operation is ultimately effective- is something which is questionable.

The question from the political aspect is what the Kurdish leadership does about Kurdish fighters/terrorists who want a larger Kurdish state- or these that hold that warfare is only effective political action to resolve "problems" of Kurdish people in other countries [such as Turkey]. And since Kurdish leaders are part of Iraqi leadership, non-Kurd Iraqi leadership will also be involved.

Basically if Turkey interests is to damage/weaken/slow terrorist activity in their country by attacking operational bases in Iraq- then they could be successful. If they want to stop it- that won't be successful. Ultimately it is always in the end, about a political solution.

"WASHINGTON (CNN) -- The United States has designated a Kurdish militant group in Turkey as a terrorist organization, the State Department said.
The group, the Kurdistan Freedom Falcons, known as TAK, is affiliated with the Kurdistan Workers' Party, or PKK -- the separatist Kurdish group that has been attacking Turkey from inside northern Iraq."
http://www.cnn.com/2008/POLITICS/01/11/us.turkey/index.html