Question on commutativity of operators

[mnb96]

Hello,

I have an unitary operator f, and another binary linear operator g.
I would like to find out a necessary and/or sufficient condition on f for the following to hold:

$$f(g(a,b)) = g(f(a),f(b))$$

Is this always valid when f is linear?

 

[chiro]

Hello,

I have an unitary operator f, and another binary linear operator g.
I would like to find out a necessary and/or sufficient condition on f for the following to hold:

$$f(g(a,b)) = g(f(a),f(b))$$

Is this always valid when f is linear?

Hey mnb96.

As a start can you show what the definition of linearity is in your context?

If they are linear operators, they are going to something like matrices, and from this you can probably figure out the exact constraints for your relationship to hold.

So given this, what can you do with this information?

 

[mnb96]

Hi Chiro,

when I said that g is linear I meant that the following holds:

1) $g(\lambda a + \mu c, b)=\lambda g(a,b) + \mu g(c,b)$

2) $g(a, \lambda b + \mu c)=\lambda g(a,b) + \mu g(a,c)$

In my case, λ and μ are scalars, a,b,c are functions, and to be even more specific g is the convolution operator. So I essentially want to find the necessary and/or sufficient condition(s) on f to have:

$f(a * b) = f(a) * f(b)$

Now I am actually thinking that perhaps the original question could not be answered if we don't know the exact nature of the operator g (=in our case convolution). Just saying that "g is linear" is probably not sufficient to deduce conditions for which f(a*b)=f(a)*f(b) holds, but I might be wrong.