positive operator

A positive operator is the operator analogue of a non-negative number: it never sends a vector to point 'backward' relative to itself. Geometrically, the operator and the vector always make a non-obtuse angle, so the inner product <T x, x> is never negative. These are exactly the operators that have square roots and that arise as T* T for any T.

On a complex Hilbert space, a bounded operator T is positive (positive semi-definite) if it is self-adjoint and <T x, x> >= 0 for all x. (On a complex space the condition <T x, x> >= 0 for all x already forces T to be self-adjoint, so the self-adjointness is automatic there; on a real space it must be assumed.) Equivalently, T is positive iff it is self-adjoint with spectrum contained in [0, infinity).

Every positive operator has a unique positive square root, a positive operator S with S^2 = T, written T^{1/2}. Conversely, for any bounded operator A, the operator A* A is always positive; its square root |A| = (A* A)^{1/2} is the 'absolute value' used in the polar decomposition A = U|A|. Be careful: 'positive' here means >= 0 (semi-definite); strictly positive / positive definite adds <T x, x> > 0 for x nonzero, which is a stronger requirement.