How can I better understand manipulating "operators" in mathematical relations?

Question

Sometimes, (especially in physics), it's common to see mathematical relations manipulated and/or derived by separating "operators" from the things they "act on." I can usually keep up with and follow derivations when reading along in the book, but it bothers me that I don't really understand how it's "justified" to do that.

A classic example would be:

$$ \bigtriangledown = \left<\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right>$$

We can "apply" the operator in different ways, and I can see intuitively how it works - it appears that "multiplying" the operator by a "thing" is what "applies". This is how we can write:

$$ \bigtriangledown \cdot F = \frac{\partial U}{\partial x} + \frac{\partial V}{\partial y} + \frac{\partial W}{\partial z}$$

if

$$ F = U\hat{x} + V\hat{y} + W\hat{z}$$

And similarly with $\bigtriangledown \times$ to define the curl.

But what exactly are we allowed to do with operators and what are we not allowed to do? Is there a name for this kind of treatment? What "are" operators and what rules do they obey? For example, it seems obvious that the "square root" of an operator wouldn't make any sense. Furthermore, it seems to be a given that "squaring" a derivative operator turns it into a "second derivative" operator.

How can I learn more about this?

Answer 1

You may consider an operator as a function of functions, i. e., an operator is a function which arguments are functions itself. For example, if $f\in C^0([0,1)]$ you may declare a function $T\colon C^0([0,1])\to R$ which arguments are functions via $$T(f):=\int_0^1 f(x)\,dx.$$ So the term “operator” reflects that we're dealing with a function of functions as we call a function a “map” in a geometrical context.

Answer 2

An operator is a function. That's all. It's a rule that takes inputs (usually numbers) and produces outputs (usually numbers).

We are allowed to do at least one thing with an function: apply it. If we have two functions $f$ and $g$, we can always define a new function $f\circ g$ by applying $g$ and then $f$, in other words, $(f\circ g)(x)=f(g(x))$. This is called function composition.

Since we can compose two functions, we can compose a function with itself. In that case, we use the notation $f^2$ for $f\circ f$, and $f^n$ for the obvious generalization. That's all that's going on when we "square" an operator.

When the outputs of two functions are things that can be added together, for instance if both functions have numbers as outputs, then we can define their sum $f+g$ as a function obtained by applying $f$ and $g$ and then summing the results: $(f+g)(x)=f(x)+g(x)$. We can do the same with multiplication and division, and so on. If you think of $+$ and $\times$ as functions themselves (accepting two inputs) then this is really just a special case of composition.

So there's nothing magical going on, all we ever do with operators (functions) is apply them to one another. Things like the Laplacian or Curl are nothing more than multiple layers of composed functions.

As to why differential operators, those are simply functions that map functions to functions. The first derivative, for instance, is something that takes a function and maps it to its derivative (another function). We can just compose that with itself to obtain the second derivative: composing it with itself means applying it twice.

As a final note, you mention square roots of functions: those actually do exist, you just need to look for a function which when applied twice in a row has the same effect as the function whose square root you want.