Twenty-five years ago, when the daughter of Arnold Schoenberg was working through his archive for a book project, she came across an empty picture frame. It was missing what the Schoenberg family calls one of the composer’s most precious possessions: a signed picture of Gustav Mahler with a musical quotation from Mahler’s Symphony No. 2.

Now that picture appears to have re-emerged. A Los Angeles man has contacted the Schoenberg family and offered to sell it back for $350,000, saying the photograph was a gift to his grandfather from a composer in Schoenberg’s circle. Schoenberg family members say they doubt the gift story, suspect that the photo was at some point stolen by someone, and have demanded its return. They suggested that they might take legal action....