Positive element - Definition

In functional analysis, a hermitian element A of a C*-algebra is a positive element if its spectrum consists of positive real numbers. Equivalently, A has a hermitian square root, that is an element B of the C*-algebra satisfying B*=B and B²=A.

If A is a bounded linear operator on a Hilbert space H, then this notion coincides with the condition that

<math>\langle Ax,x\rangle<math>

be positive for every vector x in H.