Expectation Value of a Dynamical Variable In quantum mechanics, we generally take about "expectation values of dynamical variables". However, by the postulates of quantum mechanics, every dynamical variable in quantum theory is represented by its corresponding operator.
Is it therefore, incorrect to talk about "expectation value of an operator" (rather than "expectation value a dynamical variable")? Is is just semantics or is something more going on here?
In other words is it incorrect to write:
$$<\widehat{A} > =\int  \psi \widehat{A}\psi^{*}\mathbf{ d^{3}r}$$
instead of
$$<A > =\int  \psi \widehat{A}\psi^{*}\mathbf{ d^{3}r}?$$
 A: Remember, operators are nothing but maps. Expectation value of an operator is pretty much defined (I guess) in general operator theory. It just turns out that in QM (Hermitian) operators correspond to dynamical variables. In general you can also calculate expectation values of operators like $L_+$ and $a^{\dagger}$ etc., which don't have any dynamical variables associated !! 
On a more general note, if you want to get a representation for the operator (given some basis, which could be the eigenset of an operator), the expectation values would be the diagonal elements of the representation.
It is related to Linear algebra, This might throw some light on Operator theory. Linear operators (bounded of course) are maps defined in LVS that take one vector to another vector in the same LVS.
$$ \hat O : V \rightarrow  V$$
$$  \hat O \left|\psi\right> = \left|\phi\right> $$
Then you have the inner product defined on the LVS, that takes two vectors to one complex number :
$$ \left<\psi|\phi\right> : V^* \times V \rightarrow \mathbb{C} $$
Using these two, one can define the expectation value of an operator to be, 
$$ <\hat O>  = \left<\psi\right|\; \big(\hat O\left|\psi\right>\big) = \left<\psi|\phi\right> \in \mathbb{C} $$
A: Is it therefore, incorrect to talk about "expectation value of an operator"?
Yes, because when you write those integrals you're asking for the average of a dynamical variable $A$ whose associated operator is $\hat A$. The point is that you're asking for a number , e.g. the average position of the electron in a gaussian wavepaket. Then, when you want to calculate this expectation value $\langle x \rangle$, the associated operator shows up in the integrals. And it's not only semantics: what's the use of asking, let's say, "hey, what's the average of the Hamilton operator?" ?
This would mean that you had a bunch of Hamilton operators and you wanted to know the average of those operators ... This makes no sense. ^^ 
You could ask for the average energy of a system: then you would use the Hamilton operator in the integrals you wrote.
So the answer to your second question (Is it incorrect to write ...) is simply yes.
EDIT:
(This stuff is confusing...)
I just opened my Shankar and came across this:

See equation 7.3.2.: he is looking for the mean energy $\langle E \rangle$. So according to my reasoning, he writes $\langle \Psi|H|\Psi \rangle$. But then he writes $...=\langle H \rangle$, which confused me a lot. After some thinking I came up with the solution: here, he treats $H$ as an observable. And the way he treats the Hamiltonian the next section proves this: the momentum and position are treated like normal observables and not like operators.
Then, the next page, he writes this:

So here he uses $\langle H \rangle$ as a synonym for the mean energy.
