For a problem I'm working on, I need the following implication. $A,B$ are two closed densely defined operators on a Hilbert space $H$. I'll be a bit vague about the setting, add assumptions at will as necessary (the operators $A,B$ can be quite nice but unfortunately not selfadjoint). Is it true that $$ |(Av,Bv)|\le c \|v\|^2 \quad\forall v\in D \qquad\text{implies}\qquad |(A^*v,B^*v)|\le c' \|v\|^2 \quad\forall v\in D' $$ (where $D,D'$ are some suitable dense subspaces)? If $A=B$, or if $A^*$ and $B$ commute in a suitable sense, or more generally if the commutator $[A^*,B]$ is densely defined and bounded then the statement seems true. Do you see any way to relax these conditions?
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asked Mar 2, 2017 at 10:59
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