Hermitian operator: Eigenfunction conjugate for the same eigenvalue

Question

In section 17.3.4 of the textbook Mathematical Methods for Physics and Engineering, the author claims that since \begin{align*} \mathcal{L}y_i = \lambda_i \rho y_i \end{align*} for an Hermitian linear operator $\mathcal{L}$ and any eigenvalue-eigenfunction pair $\lambda_i, y_i$, if we apply complex conjugate on both sides, then we have \begin{align*} \mathcal{L}y_i^* = \lambda_i^* \rho y_i^* = \lambda_i \rho y_i^* \end{align*} as the eigenvalue has to be real for Hermitian operator. Then $y_i$ and $y_i^*$ both correspond to the same eigenvalue. But the problem is that when applying complex conjugate on both sides, isn't $\mathcal{L}$ turned to $\mathcal{L}^*$ as well? So the argument would not hold. Am I missing something here?