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I'm reading an article and there is a differential equation cast as an operator equation: $$f_n-d_x^2 f_n = \lambda f$$ where ##f_n = \partial_n f##, which is derivative of ##f## normal to a given parameterized curve. The author casts the ODE as $$B[f_n] = \lambda A[f_n]:\\ B[f_n] \equiv f_n-d_x^2 f_n,\\A[f_n] \equiv f.$$

So if we simply take ##A^{-1}## and ##B^{-1}## to both sides of the operator equation we have $$A^{-1}[f_n] = \lambda B^{-1}[f_n]$$

However, if we rewrite the operator equation as $$B[f_n] = \lambda f \implies\\ f_n = \lambda B^{-1}f.$$

Recall ##A[f_n] \equiv f \implies A^{-1}[f] \equiv f_n##. Then the rewritten operator equation is $$A^{-1}[f] = \lambda B^{-1}[f].$$

Notice one inverse equation operates on ##f## and the other on ##f_n##. How can I tell which is correct? (I know it is ##f##, just not sure where the logic went wrong).

PS sorry, I can't figure out how to label equations here.