Aleksandrov–Clark measure

The original construction of Clark relates to one-dimensional perturbations of compressed shift operators on subspaces of the Hardy space:

${\displaystyle H^{2}(\mathbb {D} ,\mathbb {C} ).}$

By virtue of Beurling's theorem, any shift-invariant subspace of this space is of the form

${\displaystyle \theta H^{2}(\mathbb {D} ,\mathbb {C} ),}$

where ${\displaystyle \theta }$ is an inner function. As such, any invariant subspace of the adjoint of the shift is of the form

${\displaystyle K_{\theta }=\left(\theta H^{2}(\mathbb {D} ,\mathbb {C} )\right)^{\perp }.}$

We now define ${\displaystyle S_{\theta }}$ to be the shift operator compressed to ${\displaystyle K_{\theta }}$, that is

${\displaystyle S_{\theta }=P_{K_{\theta }}S|_{K_{\theta }}.}$

Clark noticed that all the one-dimensional perturbations of ${\displaystyle S_{\theta }}$, which were also unitary maps, were of the form

${\displaystyle U_{\alpha }(f)=S_{\theta }(f)+\alpha \left\langle f,{\frac {\theta }{z}}\right\rangle ,}$

and related each such map to a measure, ${\displaystyle \sigma _{\alpha }}$ on the unit circle, via the Spectral theorem. This collection of measures, one for each ${\displaystyle \alpha }$ on the unit circle ${\displaystyle ^{\mathbb {T} }}$, is then called the collection of AC measures associated with ${\displaystyle \theta }$.

The collection of measures may also be constructed for any analytic function (that is, not necessarily an inner function). Given an analytic self map, ${\displaystyle \phi }$, of the unit disc, ${\displaystyle ^{\mathbb {D} }}$, we can construct a collection of functions, ${\displaystyle u_{\alpha }}$, given by

${\displaystyle u_{\alpha }(z)=\Re \left({\frac {\alpha +\varphi (z)}{\alpha -\varphi (z)}}\right),}$

one for each ${\displaystyle ^{\alpha \in \mathbb {T} }}$. Each of these functions is positive and harmonic, so by Herglotz' Theorem each is the Poisson integral of some positive measure ${\displaystyle \mu _{\alpha }}$ on ${\displaystyle ^{\mathbb {T} }}$. This collection is the set of AC measures associated with ${\displaystyle \varphi }$. It can be shown that the two definitions coincide for inner functions.

• Douglas Clark, One-dimensional perturbations of restricted shifts, J. Analyse Math., 1972, vol 25, pp 169–191.
• E. Saksman, An elementary introduction to Clark measures, in Topics in complex analysis and operator theory, Univ. Málaga, Málaga, 2007, pp. 85–136.