What is an OPERATOR?
I try to understand operator-valued kernels. For this purpose, first want to know what is an operator. I can see the definition of operator here, but I do not quit get it. Can anyone explain it in simple words, maybe with examples?
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$\begingroup$An operator is a special kind of function. The simplest functions take a number as an input and give a number as an output. Operators take a function as an input and give a function as an output.
As an example, consider $\Omega$, an operator on the set of functions $\mathbb{R} \to \mathbb{R}.$ We can define $\Omega(f) := f + 1$. The operator $\Omega$ takes the function $x \mapsto f(x)$ as an input and gives $x \mapsto f(x)+1$ as its output function.
Another, well known, linear operator is differentiation. In this example:
$$\Omega(f) := \frac{df}{dx} \, . $$
It is a linear operator because $\Omega(\lambda f+\mu g) = \lambda\Omega(f) + \mu\Omega(g).$
Functions are increadibly general objects, so operators are even more so. Operators are functions on functions. If you're still stuck then I recommend you spend more time thinking about functions.
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