April 25, 2024
Let X be a compact subset of the complex plane \bm{C}. Define \|f\|_\infty = \sup_{x\in X} |f(x)| and \|f\|_d = \sup_{z\not=w}|f(z) - f(w)|/|z - w|\}. Define the Lipschitz norm of f\colon X\to\bm{C} by \|f\| = \|f\|_\infty + \|f\|_d and let \operatorname{Lip}(X) be the set of functions with finite Lipschitz norm. \operatorname{Lip}(X) is a Banach algebra under this norm.
For a real measure \mu\in M(X) define \|\mu\|_{KR} = \int_{\mu_0\in M_0(X)} \|\mu_0\| + \|\mu - \mu_0\|_{TV} where M_0(X) are measures \mu_0 with \mu_0(X) = 0 and TV is the total variation. Let AE(X) be the space of real measures under this norm.
Theorem. AE(X)^*\cong \operatorname{Lip}(X)
Define \phi\colon\mathcal{C}_1\to AE(X) by \phi(L)(z) = \operatorname{tr}(z - T)^{-1}L
Fix T\in\mathcal{B}(\mathcal{H}).
\Phi\colon R(X)\to\mathcal{B}(\mathcal{H}). If f\in H(X) then f(z) = (1/2\pi i)\int_\gamma f(w)(w - z)^{-1}\,dw.
f(T) = (1/2\pi i)\int_\gamma f(w)(w - T)^{-1}\,dw.
\langle f(T), L\rangle = (1/2\pi i)\int_\gamma f(w)\operatorname{tr}(w - T)^{-1}L\,dw.
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\mathcal{B}(\mathcal{H})\to M(X)/R(X)^\perp