This article explains if it is possible to know who deleted a ticket.
A ticket may get deleted accidentally by a Staff member. Once a ticket is deleted from the help desk, it is also removed from the database. You can always check the name of the staff member who deleted a ticket from the Activity Log.
Activity Log logs every activity performed in the admin control panel and the staff control panel e.g. inserting new staff user, editing new email queue, deleting tickets, etc. These logs give an insight into the activities in the helpdesk and can be found at Admin CP > Logs > Activity log