Dirichlet algebra

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In mathematics, a Dirichlet algebra is a particular type of algebra of rational functions, associated to a subset X of the complex plane. It lies inside C(X), the algebra of bounded continuous functions on X

Let $\mathcal{R}(X)$ be the set of all rational functions that are continuous on $X$ ; in other words functions that have no poles in $X$ . Then

$\mathcal{S} = \mathcal{R}(X) + \bar{\mathcal{R}(X)}$

is a *-subalgebra of $C (X)$ , and of $C\left(\partial X\right)$ . If $\mathcal{S}$ is dense in $C\left(\partial X\right)$ , we say $\mathcal{R}(X)$ is a Dirichlet algebra.

It can be shown that if an operator $T$ has $X$ as a spectral set, and $\mathcal{R}(X)$ is a Dirichlet algebra, then $T$ has a normal boundary dilation. This generalises Sz.-Nagy's dilation theorem, which can be seen as a consequence of this by letting

$X=\mathbb{D}$ .