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 $x$.
letter-openingcurlybracket-function-closingcurlybracket denote an integral transform, such as the common Fourier Trafo notation $\mathcal F\left\{x(t)\right\}(f)$; the $(t)$ and $(f)$ are often even left out, because they just explicitly state the free variable's symbol before and after transformation, so $\mathcal F\left\{x\right\}=X(f)$ is pretty common.
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– Marcus Müller
Oct 14 '17 at 12:36
Mathematically, an electrical circuit is as an operator, i.e., a function that takes a function and returns another function. Let this operator be denoted by $\mathcal T$, let $x : \mathbb R \to \mathbb R$ be the input signal and let $y := \mathcal T (x)$ be the output signal. Let $\mathcal D_{t_0}$ be the delay operator that delays its input by $t_0 > 0$.
Translating the equation on your book to this language, we obtain
$$(\mathcal T \circ \mathcal D_{t_0}) (x) = \mathcal D_{t_0} (y) = (\mathcal D_{t_0} \circ \mathcal T) (x)$$
which holds for all input signals $x$. Hence, $\mathcal T \circ \mathcal D_{t_0} = \mathcal D_{t_0} \circ \mathcal T$, i.e., operators $\mathcal T$ and $\mathcal D_{t_0}$ commute. Thus, the electrical circuit corresponding to operator $\mathcal T$ is time-invariant.
Since the word did not appear in previous answers, I would suggest the meaning of "transfer" function. Systems theory is globally about relationship between inputs, outputs, and some process in-between, sometimes called an input–process–output (IPO) model:
The input–process–output (IPO) model, or input-process-output pattern, is a widely used approach in systems analysis ... resources, money or information, transformed into outputs, such as consumables, services, new information or money.
So the idea of a transform/transformation is at its core. More specifically in engineering, one sometimes uses the notion of transfer function:
In engineering, a transfer function (also known as system function or network function of an electronic or control system component, is a mathematical function giving the corresponding output value for each possible value of the input to the device
which I believe is appropriate too in your concept. See for instance SE How to interpret the notation of a transfer function. Aside, $T$ is a common notation for an integral transform:
In mathematics, an integral transform is any transform $T$ of the following form: $$ (Tf)(u)=\int_{t_{1}}^{t_{2}}K(t,u)\,f(t)\,dt$$
