Positive operator

Summary

A positive operator is a bounded self-adjoint operator $A$ on a Hilbert space with $\ip{Ax}{x} \ge 0$ for every vector $x$. It is the operator version of a positive semidefinite matrix and of a nonnegative real number. Positive operators define an order $A \le B$ on self-adjoint operators, they have square roots, and every operator of the form $T^*T$ is positive.