The Exercise

A positive operator $T$ on a complex inner product space $V$ is a linear operator on $V$ whose expectation value in every vector, normalized or not, is a real, nonnegative number: $$\langle v|T|v\rangle \ge 0 \qquad \text{for every } |v\rangle \in V.$$(1) Density operators and projectors are positive. The exercise asks to show that every positive operator $T$ is Hermitian, that is, equal to its adjoint.

Hermitian and Symmetric Operators

The adjoint $T^\dagger$ of a linear operator $T$ on a finite-dimensional $V$ is the linear operator on $V$ with $$\langle w|T^\dagger|v\rangle = \langle v|T|w\rangle^* \qquad \text{for all } |v\rangle, |w\rangle \in V,$$(2) and a Hermitian, or self-adjoint, operator $T$ on a finite-dimensional $V$ is a linear operator equal to its adjoint. A symmetric operator $T$ on $V$, of any dimension, is a linear operator whose matrix element between two vectors is the complex conjugate of the one with the two vectors exchanged.

The Claim

What Is Checked

Both conclusions are checked for every positive operator on a complex inner product space, the first when the space is finite-dimensional.