What is the meaning of this notation? I have seen this notation in the book Schaums-Outline a couple of times now and I can't quite understand it. I'd very much appreciate if someone explained what it means. I am thinking that it means Transform, as in the input goes in the system and transforms? Thank you very much!
It just means "the transformation that turns $x$ into $y$."

You might also see $\mathbf{T}^{-1}$ which means the inverse: turning $y$ into $x$.