Transfer function of a frequency shifting system There is a system which shifts frequencies of input by $-F_c$ such that:



  $$Y(S) = X(S).H(S)$$


But $X(S)$ has value zero from $0$ to $F_c$. 

I am confused on how the product of $X(S)$ and $H(S)$ becomes a positive value in $Y(S)$ in that frequency range, for any $H(S)$? 

How transfer function of the system $H(S)$ will look like in the frequency domain?
The system you're looking for cannot be described by a transfer function because it is a time-varying system. Only linear time-invariant (LTI) system can be fully characterized by a transfer function. However, there is no LTI system that can shift frequencies. The output of a (stable) LTI system can only have frequency components that are already present in the input signal.

Frequency shifts are usually achieved by modulation, i.e., by multiplying the input signal with a sinusoid or with a complex exponential. Such a system is linear but not time-invariant.