Kraus decomposition for non trace preserving operation: shouldn't we have $0 \leq \sum_k E_k^{\dagger} E_k \leq I$

Question

In N&Chuang, on page 368 is written the following theorem:

The map $\mathcal{E}$ satisfies axioms A1,A2,A3 if and only if $$\mathcal{E}(\rho)=\sum_k E_k \rho E_k^{\dagger}$$ Where $\sum_k E_k^{\dagger} E_k \leq I$

The axiom A2 is convex linearity, the axiom A3 is CP, the axiom A1 is:

Axiom A1: $0 \leq Tr(\mathcal{E}(\rho)) \leq 1$

Shouldn't be added in the theorem: $\sum_k E_k^{\dagger} E_k \geq 0$ as well to ensure the fact the trace can never be negative ? So in the end we would have:

$$0 \leq \sum_k E_k^{\dagger} E_k \leq I$$

Answer 1

It's true for any matrix $A$ that $A^\dagger A\ge 0$. It's because $(A^\dagger A v,v)=(Av, Av)$, where $(,)$ is the inner product and $v$ is any vector.