PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.Existence.StoneReconstruction

Let $H$ be a complex Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. This definition constructs the **spectral measure** $\mu$ on the real line $\mathbb{R}$ associated with the self-adjoint closure $\bar{T}$ of the candidate Stone generator $T$ (where $T\psi = -i \frac{d}{dt} U(t)\psi \big|_{t=0}$). The measure is obtained by applying the Cayley-transform spectral measure construction to the essential self-adjoint closure of the candidate generator.

Let $H$ be a complex Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. Let $T$ be the candidate Stone generator of $U$, defined as $T\psi = -i \left. \frac{d}{dt} U(t)\psi \right|_{t=0}$ for $\psi$ in the domain where the orbit is differentiable. Then the closure of $T$, denoted $\bar{T}$, is a self-adjoint operator that satisfies the domain-aware spectral theorem with respect to the spectral measure $E$ (constructed via the Cayley transform). That is, $\bar{T}$ is the operator associated with the spectral integral of the identity function $\lambda \mapsto \lambda$ over the real line $\mathbb{R}$ with respect to $E$.

Let $H$ be a complex Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. This definition constructs the unitary operator $V(t) = e^{it\bar{T}}$ for any $t \in \mathbb{R}$, where $\bar{T}$ is the closure of the candidate Stone generator $T$ of the group $U$. The operator is defined using the spectral theorem as: $$V(t) = \int_{\mathbb{R}} e^{it\lambda} \, dE(\lambda)$$ where $E$ is the spectral measure associated with the self-adjoint operator $\bar{T}$.

Let $H$ be a complex Hilbert space and $T$ be a symmetric, partially defined linear operator on $H$. Let $y, z : \mathbb{R} \to H$ be two curves such that for all $t \in \mathbb{R}$, the vectors $y(t)$ and $z(t)$ belong to the domain of $T$. If both curves satisfy the differential equation $$\frac{dw}{dt} = i T w(t)$$ specifically, if for all $t$, the derivative of $y$ at $t$ is $i T y(t)$ and the derivative of $z$ at $t$ is $i T z(t)$, then $y(0) = z(0)$ implies that $y(t) = z(t)$ for all $t \in \mathbb{R}$. This result utilizes the symmetry of $T$ to show that the norm of the difference between two such solutions is constant.

Let $H$ be a complex Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. Let $T$ be the candidate Stone generator of $U$ with domain $D(T) \subseteq H$, consisting of vectors $x$ such that the orbit $t \mapsto U(t)x$ is differentiable at $t=0$. Let $V(t) = e^{it\bar{T}}$ be the unitary group reconstructed from the closure $\bar{T}$ of the candidate generator via the spectral theorem. For any $x \in D(T)$ and any $t \in \mathbb{R}$, it holds that $$U(t)x = V(t)x.$$

Let $H$ be a Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. Let $D(T)$ be the domain of the candidate Stone generator $T$, consisting of all vectors $\psi \in H$ such that the orbit $t \mapsto U(t)\psi$ is differentiable at $t = 0$. Then the subspace $D(T)$ is dense in $H$.

Let $H$ be a complex Hilbert space and $U: \mathbb{R} \to \mathcal{B}(H)$ be a strongly continuous one-parameter unitary group. Let $T$ be the candidate generator of $U$ (defined on the dense domain where $t \mapsto U(t)x$ is differentiable), and let $\bar{T}$ be its closure. Then for any vector $x \in H$ and any $t \in \mathbb{R}$, the action of the group $U(t)$ on $x$ is given by $$U(t)x = e^{it\bar{T}}x$$ where $e^{it\bar{T}}$ is the unitary operator defined via the spectral theorem for the self-adjoint operator $\bar{T}$.