positive (semi)definite operator
A positive operator is a self-adjoint operator T whose quadratic form never goes negative: <T v, v> >= 0 for all v. If the inequality is strict for every nonzero v, T is positive DEFINITE; if it can hit zero, positive SEMIdefinite. Think of it as an operator that always pushes a vector at least partly back toward itself, never to the opposite side.
Several conditions are equivalent and worth knowing as one package: T is self-adjoint with all eigenvalues >= 0; <T v, v> >= 0 for all v; T = S^* S for some operator S; and T has a unique positive square root sqrt(T) (a positive operator R with R^2 = T). The square root is what lets you take honest geometric means of operators.
Self-adjointness is part of the definition, not a bonus: it guarantees <T v, v> is real so the inequality even makes sense, and it delivers the real, nonnegative spectrum. In coordinates these are exactly the positive (semi)definite Hermitian or symmetric matrices from Vol I, now seen invariantly.
They are ubiquitous because squaring produces them: any A^* A is positive semidefinite (its eigenvalues are the squared singular values of A), every Gram matrix is positive semidefinite, and every covariance matrix is positive semidefinite. Positivity is the algebraic signature of something that came from a sum of squares.
Three equivalent faces of positivity: a nonnegative quadratic form, a nonnegative spectrum, and a factorization as S^* S.
The square root is genuinely useful: it gives the polar decomposition T = U sqrt(T^* T), the operator analog of writing a complex number as (magnitude)(phase). Uniqueness needs the POSITIVE root — there are many square roots, only one nonnegative one.