How to Prove a System Is Invertible?
In general LTI System is invertible if it has neither zeros nor poles in the Fourier Domain (Its spectrum).
The way to prove it is to calculate the Fourier Transform of its Impulse Response.

The intuition is simple, if it has no zeros in the frequency domain one could calculate its inverse (Element wise inverse) in the frequency domain.

Few remarks for the practice world:


If for a certain system we know the input signals are limited to certain band (Or zones) in frequency domain it is enough the LTI system has no zeros in this zone only.
In practice we have noise hence even values which are not mathematically zero but close to 0 (Or even low relative to the gain of the LTI system, Say ~-60 [dB]) are making the system not invertible.


Look for many question in this community about Deconvolution and Inverse Problems.