Unitary Dilations

Suppose that $\mathbf{H}$ is a subspace of a Hilbert space $\mathbf{K}$, and let $P$ be the (orthogonal) projection from $\mathbf{K}$ onto $\mathbf{H}$. Each operator $B$ on $\mathbf{K}$ induces in a natural way an operator $A$ on $\mathbf{H}$ defined for each $f$ in $\mathbf{H}$ by $Af = PBf$ (equivalently, $AP = PBP$). Under these conditions the operator $A$ is called the compression of $B$ to $\mathbf{H}$, and $B$ is called a dilation of $A$ to $\mathbf{K}$.

An operator $B$ is a power dilation (sometimes called a strong dilation) of an operator $A$ if $B^{n}$ is a dilation of $A^{n}$ for $n = 1, 2, 3, \ldots$.

This is the (mean) ergodic theorem for contractions. For a unitary operator $U$ this asserts that the averages $\frac{1}{n}\sum_{j=0}^{n-1} U^{j}$ form a strongly convergent sequence; more informatively, the limit is the projection whose range is the subspace $\left\{ f : Uf = f\right\}$, i.e., the subspace of fixed points of $U$.

For a bounded complex-valued function $F$ defined on a set $M$, write $\left\| F\right\|_{M} = \sup \left\{ \left|F(\lambda )\right| : \lambda \in M\right\}$. This result is known as the von Neumann inequality. A spectral set for an operator $A$ is a set $M$ such that $\operatorname {spec} A \subset M$ and such that if $F$ is a bounded rational function on $M$ (i.e., a rational function with no poles in the closure of $M$), then $\left\| F(A)\right\| \leq \left\| F\right\|_{M}$. The von Neumann inequality is frequently restated as: the closed unit disc is a spectral set for every contraction.