Operators are mathematical representations. For our purposes, these representations will model some real, physical process in mathematical notation. This operator will operate on functions in some vector space and produces outputs that lie in some other vector space. Often times (as will be the case most of the time this term), the output vector space and the input vector space are the same. The theoretical part of mathematical physics is the development of mathematical operators that capture the physical behavior that is observed in nature. Once the operators are defined, the rest is just math. In many cases the operators are accurate only up to some limit. For example, many of the systems that we experience in optics are not strictly linear, but can be modeled as such for inputs of low enough intensity.