Self-adjoint Operators and their Products: Solving for ##AB##

[member 428835]

Homework Statement

Given two linear self-adjoint operators $A,B$, is it true $AB$ is also self-adjoint.

Homework Equations

Self adjoint implies $(A[f],g) = (f,A[g])$

The Attempt at a Solution

I'm not really sure. I'm stuck almost right away: $(AB[f],g) = (A[B[f]],g) = (B[f],Ag) = (f,BAg) = (f,ABg)$. Last equality is from linearity of $B,A$. The first few steps I think follow from self-adjoint of $A,B$.

 

[Orodruin]

The last equality requires A and B to commute. That two operators are linear is certainly no guarantee for them to commute.

 

[member 428835]

The last equality requires A and B to commute. That two operators are linear is certainly no guarantee for them to commute.

Riiiiight, good call! So what should I do to prove two operators commute? Are there any sufficient conditions?

 

[Orodruin]

Given the problem formulation, are you sure that what you want to do is to prove that A and B commute?

 

[member 428835]

Given the problem formulation, are you sure that what you want to do is to prove that A and B commute?

I guess I'm mostly interested to know when two linear operators are able to commute. Unless there is a better way to know if $AB$ is self adjoint (I realize I had the question stem all bold, so I corrected it).

 

[Orodruin]

The point is that you have essentially shown that AB is self adjoint if A and B commute. So in order to answer the question ”is AB self adjoint?” you must ask yourself ”is it true that A and B always commute”. If you can find a single counter example, then you will have shown it to not be the case.

 

[member 428835]

The point is that you have essentially shown that AB is self adjoint if A and B commute. So in order to answer the question ”is AB self adjoint?” you must ask yourself ”is it true that A and B always commute”. If you can find a single counter example, then you will have shown it to not be the case.

Well both operators are linear differential operators, so they should commute then right?

 

[Orodruin]

Well both operators are linear differential operators, so they should commute then right?

No.

 

[member 428835]

No.

Thanks, I see what I'm looking for now! I really appreciate the help!

 

[member 428835]

No.

I do have a related question. Suppose we have a functional defined as $$J[f] = (A[f],f)$$ and then I want to find the stationary points of the functional, so that $\delta J = (\delta A[f],f) = (A[\delta f],f)$. Is the last equality true ($A$ is linear as mentioned above)?