In the first wave on Sunday, the Israeli air assault targeted training camps, police stations, and a Hamas intelligence headquarters. Despite the urging of colleagues and opposition politicians, Prime Minister Olmert is not talking about regime change in Gaza.
"The operation in the Gaza Strip is designed, first and foremost, to bring about an improvement in the security reality for the residents of the south of the country," said Olmert over the weekend.
On Sunday, Hamas rockets landed near Ashdod, the largest city in southern Israel. The city is 23 miles from Gaza. No serious injuries were reported, but the attack raises concerns that more Israeli cities may be within range of Hamas rockets.
With the conflict spilling over to neighboring countries, that goal may become more difficult. Arab satellite television news broadcast images of crowds of Gazans overrunning Egyptian security posts at the border with Gaza. Along the Lebanon border, the attacks have stirred concern about solidarity rocket strikes from Hezbollah.
Israeli jets flew low-level sorties over southern Lebanon Sunday morning, a muscle-flexing gesture. The militant Shiite Hezbollah has led calls of condemnation in Lebanon, declaring the attack on Gaza an "Israeli war crime and represents genocide."
But analysts say Hezbollah is unlikely to open up a fresh front by attacking Israel from Lebanon. The main risk comes from isolated attacks by Palestinian militants or groups associated with Al Qaeda.
"If it's not Hezbollah, I would not rule out actions by small groups," says Timur Goksel, a university lecturer in Beirut and former senior official with the UN peacekeeping force in South Lebanon known as UNIFIL. "There are many groups that would like to show solidarity with the Palestinians in Gaza. But if something big happens, the bill will be made out to Hezbollah."