• 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##.

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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?

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

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).

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.

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?

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!

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)?