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I have a self-adjoint linear contraction $A_g$ on an $L^2$-space of the form $$A_gf=\int\gamma(f,g),$$ where $\gamma$ is Lipschitz continuous and $g$ is an a priori fixed function. Assuming $1-A_g$ is injective and $f\in\mathcal R(A_g)$, I need to minimize the quantity $\langle(1-A_g)^{-1}f,f\rangle_{L^2}$ over al choices of $g$ for this fixed $f$.

How can we solve this problem? Maybe we can use the spectral measure $E_g$ associated to $A_g$ such that $$(1-A_g)^{-1}=\int_{\sigma(A_g)}\frac1{1-\lambda}\:E_g({\rm d}\lambda)\tag1.$$ My problem with $(1)$ is that I don't know how the measure $E_g$ depends on $g$.

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  • $\begingroup$ Why is $A_g$ linear if $\gamma$ is only assumed to be Lipschitz continuous? $\endgroup$
    – Jochen Glueck
    Oct 11, 2019 at 20:49
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    $\begingroup$ @JochenGlueck You're right. I've reformulated the question, being more specific now. See mathoverflow.net/q/343651/91890. (I'll delete this post later.) $\endgroup$
    – 0xbadf00d
    Oct 12, 2019 at 4:24

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