Generally, in maritime property cases, admiralty jurisdiction extends to petitory as well as possessory suits.  A petitory suit is an action to try title to a vessel[i].  To bring a petitory suit, the plaintiff must assert legal right to the vessel.  However, in a possessory suit, right to possession of a property that has been wrongfully taken is adjudicated.  Thus, a petitory suit is based on a claim of legal right and a mere assertion of an equitable interest is not sufficient.  Whereas in possessory suit, party seeks to adjudicate his/her right to possess property wrongfully taken.

Admiralty law provides for an in rem action to assert title or a possessory right to a vessel and other maritime property.  However, admiralty courts do not have the powers of an equity court.  Hence, an admiralty court will not assume jurisdiction to enforce an equitable interest against a legal title[ii].  Also, admiralty courts cannot take cognizance of a mere equitable title of a person who is out of possession.

However, admiralty courts can take cognizance of an equitable claim of person who is in possession of the property.  Ultimately, an admiralty court can exercise jurisdiction only over rights purely arising out of maritime activity.
[i] Privilege Yachting v. Teed, 849 F. Supp. 298 (D. Del. 1994)

[ii] Commercial Banking Corp. v. One Approximately 30 Foot Motor Boat, 86 F. Supp. 618 (D.N.J. 1949)