Is there a general method to express the resolvent of an operator $A$ with $A$ appearing in the numerator? For instance, the Neumann series expansion is one approach, but it requires the operator norm of $A$ to be less than one for convergence. I am seeking a more versatile method that does not rely on such a restriction.
Furthermore, if $\lambda$ is an imaginary number without a real part, are there any specific properties of the resolvent that I can utilize?