I am working on something and have lead to a problem. I need your help!(adsbygoogle = window.adsbygoogle || []).push({});

Let A and B be self-adjoint operators acting on a finite dimensional Hilbert space. Then, the equation

A + [ B , X ] = 0,

has at least one solution X, iff Tr(A)=0.

([ B , X ] = BX - XB)

I have proved it by taking the matrix form of the operators.

But my question is about countably infinite dimensional Hilbert spaces. There the Trace condition is irrelevant. But will there always be a solution?

I would appreciate your guidance.

Thanks

Leo

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# Need help for solving an operator equation

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