But in broadening the goals of the campaign, Israel has invited growing criticism, following the initial condemnation of the abductions by Palestinian leaders and the international community.
Palestinian officials say the sweep is boosting the Islamic militants' popularity at the expense of Western-backed Palestinian President Mahmoud Abbas. They say Israeli forces have needlessly escalated tensions, targeting sites with tenuous links to Hamas while sparking protests that have left four Palestinians dead — including a teenager.
They also argue that after years of Israeli and Palestinian crackdowns, Hamas was already weak when the new sweep began.
"Israel has no real targets," said Maj. Gen. Adnan Damiri of the Palestinian security forces. "Israel is doing a show."
In Israel, some asked why Prime Minister Benjamin Netanyahu hasn't come forward with what he claims is "unequivocal" proof of Hamas involvement in the June 12 abduction of three Jewish seminary students in the West Bank.
Others warned that pushing too hard could also endanger a key Israeli interest — continued security coordination with Abbas' forces.
Netanyahu has demanded that Abbas dismantle the Hamas-backed unity government he formed earlier this month.
Abbas has said Israel has presented no proof of Hamas involvement in the kidnapping and that the unity government is following his political program.
"After the Americans and Europeans supported the unity government, Israel used this abduction to weaken and embarrass the Palestinian Authority," Damiri said.
In the international community, there were first signs of concern about possible overreach by Israel.
Israeli must avoid "punishing individuals for offenses they have not personally committed," senior U.N. official Jeffrey Feltman told the Security Council Monday.
As part of the battle for international public opinion, the parents of the missing teens spoke to U.N. Human Rights Council in Geneva on Tuesday.
Rachelle Fraenkel, mother of 16-year-old Naftali, said she was grateful for the worldwide condemnation of the kidnappings, but that "much more can be done and should be done by many" to bring the teens home.
She told the council about the missing — that her son plays guitar, another youth loves sports and the third is an amateur pastry chef.
She also shared the parents' anguish.