“We regret that our initial media release was not totally accurate, but new evidence indicates that it was not,” the police said, adding that the conclusion was based on interviews with witnesses and “critical evidentiary items.”

[A lawyer for Mr. Bradford’s family described him as a “good guy with a gun.”]

In their initial statement on Friday, the police said uniformed officers who were providing security at the mall “encountered a suspect brandishing a pistol and shot him.” It was not clear whether the officers believed Mr. Bradford fired or intended to fire before he was killed.

Mr. Bradford’s mother, April Pipkins, said in an interview on Saturday that Mr. Bradford was living with her near Birmingham where he had been raised. Mr. Bradford, who was better known as E.J., would not have been involved in the shooting, and might have been trying to protect other people in the mall, she said.

“That was not his character at all,” she said. “He loved life, and he loved people.”

He was licensed to carry a firearm, she said. Alabama generally does not prohibit people from carrying firearms in public, according to the Giffords Law Center to Prevent Gun Violence.

At an emotional news conference held by Mr. Bradford’s relatives on Sunday — one collapsed in tears partway through — his family members said they were not notified of his death by the Hoover Police Department, but instead learned of it through social media.