I Quick question regarding creation/annihilation operators

[MaestroBach]

TL;DR Summary

Given the matrix element of an annihilation operator, can we consider it to be a creation operator acting to its left?

I'm reading a textbook and it states the following:

For whatever reason latex previewing is not working for me right now.. but the way I was thinking about this was to take the complex conjugate of the entire left, act with the ensuing creation operator on |x_1 ... x_N-1> to get |x_1...x_N>, and then take the complex conjugate of <y_1...y_N | x_1....x_N>* to get back the final result.

Is that the right way to think about it mathematically? And can I just generally think about it as a creation operator acting to its left?

 

[pines-demon]

Is that the right way to think about it mathematically? And can I just generally think about it as a creation operator acting to its left?

Yes an annihilation operator for the kets can act as a creation operator on the bras. Also note that you would usually use just a Latin letter like $\hat{a}, \hat{b},\hat{c},\hat{f}$, rarely we use $\hat{\psi}$ for annihilation/creation operators.