The Symantec Corporation, a software company in Cupertino, Calif., said today that it had agreed to acquire Peter Norton Computing Inc. of Santa Monica, Calif. for Symantec common stock valued at about $70 million.

Peter Norton is a leading developer of utility software, which is used to index hard disks and to perform other management tasks on computer systems.

Under the merger, which is expected to be completed in August, Peter Norton, founder and chairman of the company bearing his name, will become a member of the board of Symantec. He will also be the company's largest shareholder, with about 30 percent of shares outstanding. Mr. Norton will continue to be active in product development.

Symantec shares rose today by $1, to $22.50, in over-the-counter trading. Peter Norton is not traded publicly. Peter Norton had been planning a public stock offering when it was approached by Symantec with a merger proposal, Mr. Norton said. ''We weren't looking to get acquired,'' he said. ''Symantec beat our door down and convinced us this makes sense.''