I am a physicist who only knows a bit of functional analysis.
In a Quantum Field Theory class, we seem to have an operator $A$ with the property
$$A = 0$$ $$\langle\psi,AB\phi\rangle\neq0$$
where $B$ is another (unbounded) operator, for certain vectors $\phi, \psi$.
It was suggested that $A=0$ only in the sense that $\langle\psi, A \phi \rangle=0 \,\forall \psi,\phi$.
In functional analysis, is it possible to have an operator with this property? If so, is there a stronger sense in which $A=0$ for which we could not have the second property? I would be very interested in an intuition for why one might not have $A=0\implies AB=0 \,\forall B$.
Suppose that $\langle\psi,A\phi\rangle=0$ for every $\phi,\psi$, replace $\psi$ by $A\phi$ you obtain $\langle \psi, A\phi\rangle=\langle A\phi,A\phi\rangle=0$. This implies $A\phi=0$ and $A=0$, the identity $\langle \psi,AB\phi\rangle\neq 0$ cannot be true since $AB\phi=0$.