The violence coincided with stepped-up efforts by the two sides to anchor a shaky cease-fire that ended Israel's devastating three-week offensive.
Hamas security officials said the 22-year-old man was killed by an Israeli tank shell fired from across the border. Dr. Moaiya Hassanain of the Gaza Health Ministry identified him as a farmer.
The Israeli military denied that tank fire or strikes of any kind were carried out around the time of the man's death, and the circumstances of the incident remained unclear.
The Israeli military said the air strikes were launched in response to rocket fire from Gaza on Sunday, but had no immediate information on a tank shelling. No injuries were reported in the air strikes.
The military did not identify the targets of the aerial attacks. But Palestinian officials said aircraft hit a Hamas security compound in southern Gaza that had already been struck three times during Israel's recently ended Gaza offensive and a rocket-launching pad in a field in northern Gaza.
Egypt has been trying to broker a long-term cease-fire between Israel and Hamas. CBS News correspondent Robert Berger reports that the key sticking point in the negotiations is Hamas' demand that border crossings into the Gaza Strip be opened. Israel says that won't happen until Hamas frees a captive Israeli soldier.
However, CBS News' George Baghdadi reports that the militant group's second in command said Sunday a deal could be within reach. (CBS News correspondent Richard Roth. (Click here to read the full report.)
"We will bring down the Hamas government," Netanyahu said recently on the campaign trail. But with rockets from Gaza still falling on Israel, he hasn't said how he would do that.
His closest opponent is Foreign Minister Tzipi Livni, billing herself as a "different" sort of leader in Israel's male and macho politics, Roth reports. But she's promising what amounts to business as usual.
But, as talks continue, the sporadic violence persists.
In addition to Israel's air strikes and the alleged shelling, an Israeli gunboat fired a shell in the direction of a Palestinian fishing boat off the northern Gaza coast, Palestinian officials said. The military said the boat had strayed beyond limits Israel has set for fishermen, and that the gunfire was meant as a deterrent.
Palestinian officials said the fishing boat was damaged but no one was injured.
Israel unilaterally ended a devastating, three-week offensive in Gaza, meant to halt years of rocket fire on southern Israeli communities, on Jan. 18. Gaza's Hamas rulers announced their own cease-fire the same day.