From Linear Algebra Done Right:

"Positive Operators

An operator T ∈ L(V ) is called positive if T is self-adjoint and ⟨T v , v ⟩ ≥ 0

for all v ∈ V. Note that if V is a complex vector space, then the condition that T be self-adjoint can be dropped from this definition (by 7.3)."

If V is a real vector space then <Tv,v>=<v,Tv> so T is automatically self-adjoint. Doesn't this mean that the condition that T is self-adjoint can be dropped whatever the situation?