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We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where $\int_D K f\,dx$ is not a contraction and $D$ is compact, what are some of the general cases in which $f=0$ is the unique solution?

Alternatively, what restrictions on $K$ does one have to put to obtain a unique solution?

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  • $\begingroup$ If you write $Tf=\int Kf\, dx$, then you want $1+T$ to have trivial kernel, so $-1$ must not be an eigenvalue of $T$, and whether or not this holds will depend on small details, so there's not a whole lot you can say in this generality. $\endgroup$
    – Christian Remling
    Commented Sep 11, 2018 at 0:56
  • $\begingroup$ Is there perhaps some interesting subclasses of small detailed cases where this is true? Again, what (nontrivial) restrictions on $K$ could one put on equations of this form to get a unique solution? $\endgroup$
    – Ning Bao
    Commented Sep 11, 2018 at 2:09

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