Contractive Containment, Stationary Dilations, and Partial Isometries: Equivalence, Properties, and Geometric Intuition

1. Preliminaries

Definition 1 (Hilbert Space Contraction). A bounded linear operator $T: H_1 \to H_2$ between Hilbert spaces is called a contraction if $$\|Tx\|_{H_2} \leq \|x\|_{H_1} \quad \forall x \in H_1$$ Equivalently, $\|T\| \leq 1$.

Definition 2 (Stationary Process). A stochastic process $\{Y(t)\}_{t \in \mathbb{R}}$ is stationary if for any finite set of time points $\{t_1,\ldots,t_n\}$ and any $h \in \mathbb{R}$, the joint distribution of $$\{Y(t_1+h),\ldots,Y(t_n+h)\}$$ is identical to that of $\{Y(t_1),\ldots,Y(t_n)\}$.

Definition 3 (Stationary Dilation). Given a non-stationary process $X(t)$, a stationary dilation is a stationary process $Y(s)$ together with a family of bounded operators $\{\phi(t,\cdot)\}_{t \in \mathbb{R}}$ such that $$X(t) = \int_{\mathbb{R}} \phi(t,s)Y(s)ds$$ where $\phi(t,s)$ is a measurable function satisfying:

1. $\|\phi(t,\cdot)\|_{\infty} \leq 1$ for all $t$
2. The map $t \mapsto \phi(t,\cdot)$ is strongly continuous

Remark. The conditions on $\phi(t,s)$ ensure that the integral is well-defined and the resulting process $X(t)$ inherits appropriate regularity properties from $Y(s)$.

2. Main Results

Proposition 1 (Properties of Scaling Function). The scaling function $\phi(t,s)$ in a stationary dilation satisfies:

1. $\|\phi(t,s)\| \leq 1$ for all $t,s \in \mathbb{R}$
2. For fixed $t$, $s \mapsto \phi(t,s)$ is measurable
3. For fixed $s$, $t \mapsto \phi(t,s)$ is continuous

Theorem 1 (Equivalence of Containment). For a non-stationary process $X(t)$ and a stationary process $Y(s)$, the following are equivalent:

1. $Y(s)$ is a stationary dilation of $X(t)$
2. There exists a contractive mapping $\Phi$ from the space generated by $Y$ to the space generated by $X$ such that $X(t) = (\Phi Y)(t)$ for all $t$

Proof.
($1 \Rightarrow 2$): Define $\Phi$ by $$(\Phi Y)(t) = \int_{\mathbb{R}} \phi(t,s)Y(s)ds$$ For any finite linear combination $\sum_i \alpha_i Y(t_i)$: $$\begin{align*} \|\Phi(\sum_i \alpha_i Y(t_i))\|^2 &= \|\sum_i \alpha_i \int_{\mathbb{R}} \phi(t_i,s)Y(s)ds\|^2 \\ &\leq \|\sum_i \alpha_i Y(t_i)\|^2 \end{align*}$$ where the inequality follows from the bound on $\|\phi(t,s)\|$ and the Cauchy-Schwarz inequality. ($2 \Rightarrow 1$): The contractive mapping $\Phi$ induces a family of operators $\phi(t,s)$ via the Kernel theorem for Hilbert spaces. The stationarity of $Y$ and the contractivity of $\Phi$ ensure that these operators satisfy the required properties.

Lemma 1 (Minimal Dilation Property). If $Y(s)$ is a minimal stationary dilation of $X(t)$, then the scaling function $\phi(t,s)$ achieves the bound $$\sup_{t,s} \|\phi(t,s)\| = 1$$

Proof.
If $\sup_{t,s} \|\phi(t,s)\| < 1$, we could construct a smaller dilation by scaling $Y(s)$, contradicting minimality.

3. Structure Theory

Theorem 2 (Sz.-Nagy Dilation). For any contraction $T$ on a Hilbert space $H$, there exists a minimal unitary dilation $U$ on a larger space $K \supseteq H$ such that: $$T^n = P_H U^n|_H \quad \forall n \geq 0$$ where $P_H$ is the orthogonal projection onto $H$.

Lemma 2 (Defect Operators). For a contraction $T$, the defect operators defined by: $$D_T = (I - T^*T)^{1/2}$$ $$D_{T^*} = (I - TT^*)^{1/2}$$ satisfy:

1. $\|D_T\| \leq 1$ and $\|D_{T^*}\| \leq 1$
2. $D_T = 0$ if and only if $T$ is an isometry
3. $D_{T^*} = 0$ if and only if $T$ is a co-isometry

4. Convergence Properties

Theorem 3 (Strong Convergence). For a contractive stationary dilation, the following limit exists in the strong operator topology: $$\lim_{n \to \infty} T^n = P_{ker(I-T^*T)}$$ where $P_{ker(I-T^*T)}$ is the orthogonal projection onto the kernel of $I-T^*T$.

Proof.
For any $x$ in the Hilbert space:

1. The sequence $\{\|T^n x\|\}$ is decreasing since $T$ is a contraction
2. It is bounded below by 0
3. Therefore, $\lim_{n \to \infty} \|T^n x\|$ exists
4. The limit operator must be the projection onto the space of vectors $x$ satisfying $\|Tx\| = \|x\|$
5. This space is precisely $ker(I-T^*T)$

Corollary 1 (Asymptotic Behavior). If $T$ is a strict contraction (i.e., $\|T\| < 1$), then $$\lim_{n \to \infty} T^n = 0$$ in the strong operator topology.

5. Partial Isometries: The Mathematical Scalpel

Definition 4 (Partial Isometry). An operator $A$ on a Hilbert space $H$ is a partial isometry if $A^*A$ is an orthogonal projection.

Remark (Geometric Intuition). A partial isometry is like a mathematical scalpel that carves out a section of space:

• It acts as a perfect rigid motion (isometry) on a specific subspace
• It completely annihilates the rest of the space

This property makes partial isometries powerful tools for selecting and transforming specific parts of a Hilbert space while cleanly disposing of the rest.

Proposition 2 (Key Properties of Partial Isometries). Let $A$ be a partial isometry. Then:

1. $A$ is an isometry when restricted to $(ker A)^\perp$
2. $A(ker A)^\perp = ran A$
3. $A^*$ is also a partial isometry
4. $AA^*A = A$ and $A^*AA^* = A^*$

Theorem 4 (Geometric Characterization). For a partial isometry $A$: $$A^*A = P_{(ker A)^\perp} \quad \text{and} \quad AA^* = P_{ran A}$$ where $P_S$ denotes the orthogonal projection onto subspace $S$.

Proof.
The action of $A$ can be decomposed as:

1. Project onto $(ker A)^\perp$ (this is $A^*A$)
2. Apply a perfect rigid motion to the projected space

This two-step process ensures $A^*A$ is the projection onto $(ker A)^\perp$.

Remark (The "Not So Partial" Nature). Despite the name, there's nothing incomplete about a partial isometry. It performs a complete operation:

• It's a full isometry on its initial space ($(ker A)^\perp$)
• It perfectly maps this initial space onto its final space ($ran A$)
• It precisely annihilates everything else

This makes partial isometries fundamental building blocks in operator theory, crucial in polar decompositions, dimension theory of von Neumann algebras, and quantum mechanics.