Yes, in 1592, William Shakespeare was a well known dramatist. In London, 1589-91 Shakespeares’ first plays were presented. Shakespeares’ first reference in the London literary world was written in 1592, when another dramatist Robert Greene described him as “an upstart crow”.
Despite the critical remark he had received, Shakespeare’s literary reputation continued to grow. Shakespeare further established himself as a professional play-wright and actor after he joined the Lord Chamberlain’s Men, which was an acting company which performed at theaters in London.