Let $f:X\times Y \rightarrow \mathbb{R}$. I want to simplify an operator to denote the following functional transformation: $$f(x,y) - \int_{Y}f(x,y)g(y)\,dy, $$ where $g$ is the pdf of $Y$. So I define operator $T$ $$T(f)(x,y) = f(x,y) - \int_{Y}f(x,y)g(y)\,dy. $$
For function $f$, I do not think there is a problem with how I use the operator notation, but I want to use the same notation to simplify $$\frac{f(x,y)}{f(c,y)} - \int_{Y}\frac{f(x,y)}{f(c,y)}g(y)\,dy, $$ where $c$ is some fixed constant. I heard that it is mathematically incorrect to use $$T\left(\frac{f(x,y)}{f(c,y)}\right).$$
Could there be any better way to use the operator $T$ for ${f(x,y)}/{f(c,y)}$?