George J. Mitchell, the new U.S. envoy to the Middle East, arrived in Israel on Wednesday to begin testing his axiom that there's no such thing as a conflict that cannot be ended.

Yet even as Israeli and Palestinian leaders offered ideas on how the Obama administration can help bring about peace, the prevailing mood on both sides was that their decades-old fight had become almost hopelessly deadlocked.

Two obstacles were evident as Mitchell met with Prime Minister Ehud Olmert and other Israeli leaders. Hours before his arrival, Israeli warplanes pounded weapons-smuggling tunnels in the Gaza Strip, striking back for a bombing Tuesday that killed an Israeli soldier in the first deadly breach of a tentative cease-fire between Israel and the militant group Hamas. Israel conducted another air raid early today, bombing what it called a weapons factory after militants fired a rocket into Israel.

And a survey released by the Israeli advocacy group Peace Now detailed a major expansion of Jewish settlements in the West Bank over the last year, a trend that moderate Palestinian leaders called an impediment to the creation of a Palestinian state.

Militant attacks on Israel and settlement growth on land claimed by Palestinians have stymied peace efforts for years. An American "road map" calling for their mutual cessation, a formula promoted by Mitchell during an earlier foray into Middle East diplomacy, has been the accepted starting point for statehood talks since 2003.

Mitchell's new mission is viewed as a more vigorous effort to enforce that trade-off. After lunch with Olmert, the envoy said the United States would "sustain an active commitment for reaching the goal of two states living side by side in peace and security."

His initial focus is Hamas-ruled Gaza, where years of rocket fire into Israel led to a blockade by Israel and Egypt, followed by a 22-day Israeli assault that killed nearly 1,300 Palestinians.

Shoring up last week's cease-fire is of "critical importance," Mitchell said, adding that a longer-term truce should be based on an end to arms smuggling and the lifting of the blockade.

At the same time, he said, "the United States is committed to Israel's security and to its right to defend itself against legitimate threats."

Mitchell said that after his consultations in the Middle East and Europe, he will report to Obama and Secretary of State Hillary Rodham Clinton on his conclusions about the next steps.

In interviews this week, Israeli and Palestinian officials and analysts said the task was daunting and likely to frustrate the diplomat, who in 1998 helped settle the centuries-old conflict in Northern Ireland.

One reason is that the West Bank-based Palestinian Authority, which has held 14 months of fruitless U.S.-brokered talks with Israel, does not control Gaza. Nor is it likely to regain the share of power it lost there when its secular Fatah forces were ousted in 2007 by Hamas, a group that preaches Israel's destruction.

And yet without a Palestinian Authority presence in Gaza, Israel says, it will not lift the blockade.

In Israel, Olmert's tenure is over, cut short by scandals that forced early elections. The front-runner in the Feb. 10 polls, right-wing opposition leader Benjamin Netanyahu, is uninterested in pursuing Olmert's effort to resolve issues that stand in the way of Palestinian statehood -- borders, the fate of refugees and conflicting claims to Jerusalem.

Even Netanyahu's more dovish rivals for the prime minister job, Foreign Minister Tzipi Livni and Defense Minister Ehud Barak, told Mitchell on Wednesday that Israel would continue its "war on terror" in Gaza as it talked peace with the Palestinian Authority.

And yet, Palestinian analysts say, each battle in Gaza elevates Hamas' image among Palestinians as a resistance force, weakening the nonviolence message of Palestinian Authority President Mahmoud Abbas.

Abbas "is losing legitimacy" because peace talks have faltered and Israel has failed to rein in the settlements, said Khalil Shikaki, director of the West Bank-based Palestinian Center for Policy and Survey Research.

"Now it's Hamas' turn to impose conditions on him," Shikaki said.

Olmert disagreed. He told Mitchell that Israel's assault had weakened Hamas, bolstered Abbas and improved the climate for peace.

In private, Israeli officials dismiss the peace process as futile.

"A large portion of the Israeli leadership doesn't believe in a [statehood] agreement," said a senior government analyst.

He said Abbas was not in a position to make the concessions needed to seal a statehood agreement and that Hamas would never give up enough control of Gaza to allow such a deal to work.

Mitchell's assertion, after his appointment last week, that all conflicts are soluble "doesn't work here," the analyst said. "That's such an American view. . . . I dispute the notion that if you talk long enough and raise enough proposals that you can resolve the [Israeli-Palestinian] problem."

Palestinian Authority Prime Minister Salam Fayyad, who along with Abbas is to meet with Mitchell today, said they would urge the Obama administration to take two steps to end the deadlock.

One is to force Israel to halt settlement growth. Peace Now said 1,257 new structures were built in West Bank settlements last year, compared with 800 in 2007.

The other is to encourage Hamas and Fatah to create a "government of national consensus" that could negotiate and enforce a state-to-state relationship with Israel. The Bush administration opposed such efforts in favor of a policy of isolating Hamas.

Bucking the prevailing pessimism, Fayyad said Hamas might agree to a government led by Palestinians from outside Hamas and Fatah, on condition that it hold elections within a year.

The cause of Palestinian statehood is doomed if Gaza and the West Bank remain divided, the prime minister said in an interview.

"This is my message to the world: We Palestinians . . . have our political differences. Let us find our way. . . . Sooner or later our country is going to be reunited."