$\forall x \in H$ $(x,Ax) = (x,Bx) \implies A=B$

306 Views Asked by Bumbble Comm At 29 Apr 2026 - 10:37

Let $H$ be a complex Hilbert space, let $A,B$ be bounded linear operators on $H$. I want to show $\forall x \in H$ $(x,Ax) = (x,Bx) \implies A=B$.

I can show that $\forall x,y \in H$ $(y,Ax) = (y,Bx) \implies A=B$, pretty easily.

Is there a way to show that

$\forall x \in H$ $(x,Ax) = (x,Bx)$ $\implies A=B$

implies

$\forall x,y \in H$ $(y,Ax) = (y,Bx) \implies A=B$ to finish, or another method?