I'm reading the paper P. Halmos, L. Wallen, Powers of Partial Isometries, Indiana Univ. Math. J. 19 No. 8 (1970), 657–663 (http://www.iumj.indiana.edu/docs/19054/19054.asp). And I got stuck on the proof of Lemma 2, which states that
$$\text{If } U_1 \text{and } U_2 \text{ are partial isometries, then a necessary and sufficient condition that } \\U_1U_2 \text{ be a partial isometry is that } E_1 \text{ and } F_2 \text{ commute, where } E_1={U_1}^*U_1 \text{and } F_2=U_2{U_2}^*.$$
Proof
Put $U = U_1U_2$. Assume first that $E_1$ and $F_2$ commute. Then $$UU^*U = U_1U_2({U_2}^*{U_1}^*)U_1U_2 = U_1F_2E_1U_2 = U_1E_1F_2U_2 = U_1U_2 = U.$$ Since the equation $UU^*U=U$ is characteristic if partial isometries, this proves that the condition is sufficient.
To prove necessity, assume that $UU^*U=U$. Then, as above, $U_1F_2E_1U_2=U_1U_2$, and therefore $$E_1F_2E_1F_2=({U_1}^*U_1F_2)(E_1U_2{U_2}^*)=({U_1}^*U_1)(U_2{U_2}^*)=E_1F_2,$$ so that $E_1F_2$ is idempotent. Since $||E_1F_2|| \leq 1$, it follows that $E_1F_2$ is Hermitian, and hence that $E_1F_2$ commute.
I can't understand why we can conclude that $E_1F_2$ is Hermitian given that it is idempotent and bounded.