A space of polynomials $\{f_n\}$ is given, where $f_n$ is of degree $n$. The operator $T$ in this space, satisfy the relation $$T^2(f_n)+a_nT(f_n)-f_n=0$$ where $a_n$ is a scalar depending on $f_n$.
Does this equation refer to "Operator Equation"? If yes, how can we solve it?
Can we obtain an explicit formula for the operator $T$?