Linear Operator L with Zero Matrix Elements

[sayebms]

Homework Statement

Suppose a linear operator L satisfies <A|L|A> = 0 for every state A. Show that then all matrix elements <B|L|A> = 0, and hence L = 0.

Homework Equations

$<A|L|A>=L_{AA} and <B|L|A>=L_{BA}$

The Attempt at a Solution

It seems very straight forward and I don't know how to prove it but here is what I have tried:
$<B|L|A> \to$Using resolution of Identity $\to \sum_{A} <B|A><A|L|A> \to <B|L|A>=0$
Is it right or do I need to write more.

 

[Orodruin]

It is not correct, you are summing over A and at the same time assuming that $|A\rangle$ is a fixed state (i.e., the one you started with in $\langle B|L|A\rangle$).

 

[sayebms]

since the problems says for every state A so should I write as following $<A_i|L|A_i>=0 \to$ then as before
$<B_j|L|A_i>=\sum_{i}<B_j|A_i><A_i|L|A_i>=0$
is it right now?

 

[Orodruin]

No, it is still wrong. You cannot let the state you are summing over in the completeness relation be denoted by the same as the state you have on the left-hand side. It is simply not correct.

 

[sayebms]

should I solve it without the resolution of Identity?