Mercer's Theorem For The (Noncompact) Domain [0,Infty]

Let $T_K$ be a compact self-adjoint integral covariance operator on $L^2 [0, \infty)$ $$(T_K f) (z) = \int_0^{\infty} K (z, w) f (w) dw$$ defined by kernel $K$: $$K (x, y) = \sum_{k = 0}^{\infty} \lambda_k \phi_k (x) \phi_k (y)$$ where $\{\phi_n \}_{n = 0}^{\infty}$ is a sequence of orthonormal eigenfunctions in $L^2 [0, \infty)$ and $\{\lambda_n \}_{n = 0}^{\infty}$ the corresponding eigenvalues ordered such that $$| \lambda_{n + 1} | < | \lambda_n | \forall n$$ Let $T_{K_N}$ be the truncated operator with kernel $$K_N (z, w) = \sum_{n = 0}^N \lambda_n \phi_n (z) \phi_n (w)$$ then: $$\|T_K - T_{K_N} \| \leq | \lambda_{N + 1} |$$

Proof:

Let $E_N$ be the difference $T_K - T_{K_N}$. For any $f \in L^2 [0, \infty)$: Let $f = g + h$ where $g \in \text{span} \{\phi_k \}_{k \leq N}$ and $h \in \text{span} \{\phi_k \}_{k > N}$ so that $$g (x) = \sum_{k = 0}^N \langle f, \phi_k \rangle \phi_k (x)$$ and $$h (x) = \sum_{k = N + 1}^{\infty} \langle f, \phi_k \rangle \phi_k (x)$$ where by orthogonality of $g$ and $h$ $$\langle g, h \rangle = \int_0^{\infty} g (x) h (x) dx = 0$$ we have $$\|E_N f\|^2 = \langle E_N f, E_N f \rangle = \langle E_N h, h \rangle$$ because $E_N g = 0$ by construction and since $h$ is orthogonal to the first N eigenfunctions and $$| \lambda_k | \leq | \lambda_{N + 1} | \forall k > N$$ we have $$| \langle E_N h, h \rangle | \leq | \lambda_{N + 1} | \|h\|^2 \leq | \lambda_{N + 1} | \|f\|^2$$ Therefore: $$\|E_N \| \leq | \lambda_{N + 1} |$$

Remark:

This is an extension of Mercer's theorem to unbounded domains with uniform convergence.

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Eigenfunction Expansions for Mercer Kernels

Consider an integral covariance operator with Mercer kernel \(R (s, t)\)

\(\displaystyle T f (t) = \int_0^{\infty} R (s, t) f (s) \hspace{0.17em} ds\)

The eigenfunctions satisfy the equation:

\(\displaystyle T \psi (s) = \int_0^{\infty} R (s, t) \psi (s) \hspace{0.17em} ds = \lambda \psi (t)\)

where \(\{\psi_n \}_{n = 1}^{\infty}\) are the eigenfunctions with corresponding eigenvalues \(\{\lambda_n \}_{n = 1}^{\infty}\)

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.