While Prime Minister Olmert is beginning his visit to America by stressing diplomacy and internationally agreed solutions to the new regional realities, his newly appointed defense minister, Ehud Barak, was reportedly planning a military incursion into Gaza.

Politicians and analysts in Israel have started a major debate of whether the new Gaza situation calls for a major diplomatic initiative to strengthen the West Bank-based Fatah or, alternatively, a major military incursion into Gaza to crush the nascent Hamas state there and prevent its move into the West Bank.

As if last week’s Hamas victory in Gaza and its possible ramification for Israel’s southern towns were not enough, the country yesterday experienced the first mortar attack on its northern towns from Lebanon since last summer’s war against Hezbollah. Mr. Olmert said yesterday that the attack was perpetrated by an Al Qaeda Palestinian Arab faction in Lebanon.

Launching his American visit yesterday in New York City with a Central Park morning jog and a luncheon with Secretary-General Ban, Mr. Olmert said last week’s Gaza-West Bank split was a “diplomatic opportunity” for the renewal of ties with the Palestinian president, Mahmoud Abbas, known as Abu Mazen, and for further isolation of Hamas. Mr. Olmert is expected to meet President Bush on Tuesday.

Israel will extend economic and military “support for the government of Abu Mazen,” which was sworn in yesterday, as long as that government avoids “any compromise and agreements with Hamas,” Mr. Olmert told members of the Conference of Presidents of Major American Jewish Organizations last night.

Mr. Ban stressed humanitarian concerns in Gaza, calling Hamas’s victory there “a setback in terms of humanitarian, security, and the peace process in the Middle East.”

Israel, meanwhile, is now surrounded by powerful Iranian-backed armies and militias on its northern and southern borders. Yesterday, the Sunday Times of London reported that Mr. Barak plans a 20,000-troop Gaza invasion to destroy much of Hamas’s military capability “in days.” The attack would be triggered by Hamas rocket attacks against Israel or a resumption of suicide bombings, according to the paper.

According to one participant in the luncheon at the residence of Israel’s U.N. ambassador, Dan Gillerman, however, Mr. Olmert told Mr. Ban that currently there were no plans for a military ground operation in Gaza, which Israel withdrew from in August of 2005. Israel plans no immediate retaliation in Lebanon either, Mr. Olmert is said to have added.

Yesterday was “a very disturbing day because we had these attacks from Lebanon on the Israeli township of Kiryat Shmona,” Mr. Olmert said prior to the luncheon. But rather than military retaliation, he said, “we emphasize the importance” of the role of the U.N. Interim Force in Lebanon and the Lebanese Army.

Some Israeli analysts believe, however, that Hezbollah and Hamas’s military capabilities would soon enable them to threaten not only Sderot, across the border from Gaza, and Kiryat Shmona in the North near Lebanon, but also Israel’s major cities. There is a major debate between internationalists who seek diplomatic avenues — from U.N. peacekeepers to new territorial gestures toward Palestinian Arabs and Syria — and others calling to downgrade Hamas’s capabilities militarily now.

Last week in Gaza Hamas seized reams of documents and other material belonging to the CIA and intelligence services from Britain to Egypt that have propped up Mr. Abbas’s Fatah party in the last few years. Even with reinvigorated Western and Israeli support, the utter Fatah collapse in Gaza could eventually repeat itself in the West Bank as well, these analysts argue.

Last week, Mr. Olmert floated the idea of deploying an international force in Gaza, which Mr. Ban quickly embraced. Israel’s foreign minister, Tzipi Livni, said such a force would not only be there to observe, but to fight smugglers of weapons from Egypt into Gaza. But then came yesterday’s rocket launch. Beefing UNIFIL up was meant to end such attacks.

Yesterday’s luncheon at the residence of Israel’s U.N. ambassador, Mr. Gillerman, was no longer dedicated to an international force to Gaza, now deemed unrealistic even by Mr. Olmert. Instead, Mr. Olmert and Mr. Ban explored ways for UNIFIL to do more to stop events such as yesterday’s rocket shooting, including strengthening its mandate and perhaps even extending its deployment to Syria’s border with Lebanon as well.

The veteran military analyst Ron Ben Yishai yesterday wrote on the website Y-Net that Israel’s army must first enter Gaza. Only after a significant dismantling Hamas’s military infrastructure there could Israel mobilize its allies to create a multinational force that would assure no new weapons are smuggled in.

Before leaving for America, Mr. Olmert last week quickly moved to replace defense minister Amir Peretz with the recently elected Labor leader, Mr. Barak, who is said to be seeking to distance himself from the Oslo years in which he served as Prime Minister.

Addressing recent diplomatic overtures from Syria and other Middle Eastern countries, Mr. Barak’s top rival, Benjamin Netanyahu of the Likud Party, yesterday said during a tour of the Golan Heights that unilateral departures from Lebanon and Gaza begat “Hezbollahstan and Hamastan.” These withdrawals should be repeated “neither in the Golan Heights nor in Samaria, because we have no negotiation partner,” Mr. Netanyahu said.

Politically marginalized after the Gaza disengagement in 2005, Mr. Netanyahu suffered a spectacular election defeat in 2006. But his warnings are now much more acceptable to Israelis, who give him a considerable edge in political polls over Mr. Barak, as well as over Mr. Olmert, whose numbers are in single digits.