Observables are the linear self-adjoint operators.

• $\langle\psi|A\ \phi\rangle\in \mathbb C$ is called transition amplitude.

• $\frac{|\langle\psi|A\ \phi\rangle|^2}{\Vert\psi\Vert^2\Vert\psi\Vert^2}\ge 0$ is called transition probability.

• $\langle \psi | A\ \psi \rangle\in \mathbb R$ is called expectation value.

• $\frac{\langle \psi | A\ \psi \rangle}{\Vert\psi\Vert^2}\in \mathbb R$ is called mean value.

Wikipedia: Self-adjoint operator, Observable

Self-adjoint operator, Vector space endomorphism