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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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up vote 6 down vote accepted

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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Thanks a lot. Also, can can an operator be applied to many arguments? Like $\Omega(f, \, g) = f+g $? –  user25004 Sep 29 '12 at 1:50
    
Next, what does "If the range is on the real line or in the complex plane, the mapping is usually called a functional instead" mean? Because, by the explanations above the range should be a set of functions, not numbers. –  user25004 Sep 29 '12 at 1:57
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@user25004 In the case of a functional, the input is a function while the output is a number. You can think of the numbers as being the constant functions, i.e. $f(x) = c$ for all $x$. –  Fly by Night Sep 29 '12 at 15:01
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