Aleksandrov-Clark measure

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In mathematics, Aleksandrov-Clark (AC) measures are specially constructed measures named after the two mathematicians, A. B. Aleksandrov and Douglas Clark, who discovered some of their deepest properties. The measures are also called either Aleksandrov measures, Clark measures, or occasionally spectral measures.

[edit] Construction of the measures

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

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

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

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

where $θ$ is an inner function. As such, any invariant subspace of the adjoint of the shift is of the form

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

We now define $S θ$ to be the shift operator compressed to $K θ$ , that is

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

Clark noticed that all the one-dimensional perturbations of $S θ$ , which were also unitary maps, were of the form

$U_\alpha (f) = S_\theta (f) + \alpha < f , \frac{\theta}{z} > ,$

and related each such map to a measure, $σ α$ on the unit circle, via the Spectral theorem. This collection of measures, one for each $α$ on the circle, is then called the collection of AC measures associated with $θ$ .