Let be a complex Banach space and a closed subspace with involution. Let be a continuous real trilinear map , which is symmetric complex bilinear in , and conjugate linear in . Certain algebraic postulates for are assumed, including (), where is the operator and is the set of all operators on which are Hermitian in the sense of Vidav. Such systems are called here Partial -triples.
The main result is that when the system is geometric (all vector fields () are complete in some bounded balanced domain in ), then every Hermitian operator () has a non-negative spectrum.