According to Wikipedia, an integral transform is any transform $T$ of the following form: $$(Tf)(u) = \int_{t_1}^{t_2}f(t)K(t, u)dt.$$
The inverse transform is of the following form: $$f(t) = \int_{u_1}^{u_2}(Tf)(u)K^{-1}(u, t)du.$$
What conditions must hold to have a valid integral transform and associated inverse transform?
The Wikipedia page goes on: "every integral transform is a linear operator". Therefore, we can represent an integral transform as matrix $A$ and its associated inverse transform as matrix $A'$. What properties do matrices $A$ and $A'$ have?