# Basic question about commutation in Dirac notation

In a nutshell my question is: "In the bracket notation, we can operate in the bra and in the ket with the same operator like $$\langle a|(A|b\rangle)=(\langle a|A)|b\rangle~?$$ So this operations are correct? $$A|b_i\rangle=\sqrt{b_i}|b_i-1\rangle\leftrightarrow \langle b_i| A^\dagger=(\sqrt{b_i})^*\langle b_i-1|;$$

$$A^\dagger|b_i\rangle=\sqrt{b_i+1}|b_i+1\rangle\leftrightarrow \langle b_i|A=(\sqrt{b_i+1})^*\langle b_i+1|;$$ then applying the first equation above in the Ket $$\langle b_i|A|b_i+1\rangle=\langle b_1|\big(A|b_i+1\rangle\big)=\langle b_1|\big(\sqrt{b_i+1}|b_i\rangle \big)=\sqrt{b_i+1}$$ Now applying the second relation in the Bra $$\langle b_i|A|b_i+1\rangle=\big(\langle b_i|A\big)|b_i+1\rangle= (\sqrt{b_i+1})^*\langle b_i+1|b_i+1\rangle=(\sqrt{b_i+1})^*$$ For that to be true it's required that the values $$\in \mathcal{R}$$.

We have to be careful when we use bra-ket notation to deal with non-hermitian operators. In these cases, it is more convinient to use $$(\cdot,\cdot)$$ notation.
Being $$a$$ and $$b$$ two states, and $$A$$ an operator,
$$(a,Ab)=(A^\dagger a,b),$$
which in bra-ket notation would be $$\langle a|A|b\rangle=\langle a|\Big(A|b\rangle\Big)=\Big(A^\dagger|a\rangle\Big)^\dagger|b\rangle.$$
Then, you can see that your last equation is right only if your $$A$$ was hermitian.