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:

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

Danylo Y · Accepted Answer

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

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

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