The Israeli military has verified that a senior officer was killed in a confrontation with militants on the border with Lebanon on Monday.
According to the military, personnel and aircraft eliminated two "terrorist infiltrators" who had entered Israeli territory.
They claimed to be members of the Palestinian Islamic Jihad. In retaliation for the mortar fire, the military subsequently carried out attacks in Lebanon, which resulted in the deaths of at least three Hezbollah fighters.
Tuesday, Hezbollah released a statement claiming that its militants fired a guided anti-tank missile at an Israeli military vehicle located in the Avivim district.
The military stated that there were no casualties and that a helicopter responded by attacking one of the group's observation posts.
The Palestinian militant organisation Hamas also asserted that its fighters in Lebanon had fired missiles at the western Galilee region of Israel.
The missiles were either intercepted or fell in open areas, according to the Israeli military, which responded by shelling two additional Hezbollah observation posts.
It was the third consecutive day of attacks launched from Lebanese territory, coinciding with a conflict between Israel and Hamas in Gaza.
Hezbollah is the most powerful force in Lebanon and fought a conflict with Israel in 2006. The United States has warned Hezbollah not to get involved.
The Israeli military stated that Monday's deadly violence began when a number of armed men crossed the border near the Lebanese village of Dhayra and were killed by Israeli troops backed by helicopter gunships.
According to the report, two of the militants were dead and one escaped back to Lebanon.
The Palestinian Islamic Jihad stated that the incursion was part of "Operation Al-Aqsa Flood" - the name given by the militant group Hamas to the unprecedented attack from Gaza on southern Israeli communities that began on Saturday and is alleged to have killed at least 900 Israelis.
The Israeli military also reported that its helicopters attacked Hezbollah positions inside Lebanon, including two observation posts, after two mortar bombs were launched into Israel but no casualties were reported.
After some time had passed, Hezbollah made the announcement that three of its fighters had died as a result of Israeli attacks on Lebanese cities and villages.
According to the group's statement, in response, they launched rockets and mortar shells at two Israeli military barracks. In what it termed an act of "solidarity with the Palestinian resistance," Hezbollah on Sunday launched a number of shells and rockets at Israeli military positions in the contested region of Shebaa Farms and Mount Dov.
The group, which is backed by Iran and possesses missiles with a considerable range that are capable of striking deep into Israeli territory, has vowed to intervene in the battle between Israel and Gaza if certain "red lines" are broken. According to reports, they include an invasion by Israel.
On Monday, a senior official in the United States Department of Defense made the following statement to sources: "We are deeply concerned about Hezbollah making the wrong decision and choosing to open a second front to this conflict."