I ended up in the following equation:
$J=2F^{\dagger}JF-F^{\dagger}FJ-JF^{\dagger}F$
where J is a known operator and F is unknown. I wanna specify that my operator are infinite dimensional. How do you approach the problem?
A complicated OPERATOR equation
2
$\begingroup$
linear-algebra
operator-theory
systems-of-equations
quantum-mechanics
semigroup-of-operators
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0Can you clarify what you're trying to prove or solve for? – 2017-01-11
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0I'm trying to find out the class of F operators which satisfy the equation – 2017-01-11
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0Perhaps it's worth noting that $J \mapsto 2F^{\dagger}JF-F^{\dagger}FJ-JF^{\dagger}F - J$ is a linear operator – 2017-01-12
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0Also, if we were looking instead at $$ J=2FJF^{\dagger}-F^{\dagger}FJ-JF^{\dagger}F $$ then the equation cannot hold if $J$ and $F$ are trace-class operators. – 2017-01-12
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0Hi, brilliant. May you clarify your assertions? Why the map is linear and why the equation has no solution if $J$ and $F $ are trace class operators? (I noticed you swap the operators in the first term of the RHS) – 2017-01-13
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0Take the trace and use its cyclicality. But you said *J* is given. *Is it* finite dimensional? – 2018-12-26