PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.Cayley.Certificate

Let $D$ and $E$ be bounded normal spectral data associated with an operator $U$. If the spectral measures of $D$ and $E$ are equal, then $D$ and $E$ are equal.

Let $D$ and $E$ be bounded normal spectral data associated with a bounded normal operator $U$. If for every bounded measurable function $f: \mathbb{C} \to \mathbb{C}$, the spectral integrals of $f$ with respect to the spectral measures of $D$ and $E$ are equal as bounded linear operators (i.e., $\int_{\mathbb{C}} f \, d\mu_D = \int_{\mathbb{C}} f \, d\mu_E$), then $D = E$.

For any two instances of bounded unitary spectral data $D$ and $E$ (which represent spectral measures on $\mathbb{C}$ that reconstruct a unitary operator), if their associated spectral measures are equal, then $D = E$.

Let $H$ be a Hilbert space and let $D$ and $E$ be bounded unitary spectral data on $H$. If for every bounded measurable function $f: \mathbb{C} \to \mathbb{C}$, the operator-valued integrals with respect to the spectral measures $\mu_D$ and $\mu_E$ associated with $D$ and $E$ are equal, i.e., $$\int_{\mathbb{C}} f \, d\mu_D = \int_{\mathbb{C}} f \, d\mu_E,$$ then the spectral data are equal ($D = E$).

Given a bounded unitary spectral data $D$ on a Hilbert space $\mathcal{H}$ (where the spectral measure is supported on the unit circle $\mathbb{T} \subset \mathbb{C}$ excluding $1$), this definition constructs the associated spectral measure on the real line $\mathbb{R}$. This is obtained by pulling back the spectral measure of $D$ along the Cayley transform $c: \mathbb{R} \to \mathbb{T} \setminus \{1\}$, defined by $c(x) = \frac{x - i}{x + i}$. The resulting measure is a spectral measure on $\mathbb{R}$ taking values in the operators on $\mathcal{H}$.

Let $D$ be a bounded unitary spectral data on a Hilbert space $\mathcal{H}$. Let $\mu_D$ be the spectral measure associated with $D$ (which is supported on the unit circle $\mathbb{T} \setminus \{1\} \subset \mathbb{C}$), and let $\nu_D$ be the corresponding real spectral measure on $\mathbb{R}$ obtained by pulling back $\mu_D$ along the Cayley transform $c: \mathbb{R} \to \mathbb{T} \setminus \{1\}$, $c(x) = \frac{x - i}{x + i}$. Then, the push-forward of $\nu_D$ along the Cayley transform (denoted as `cayleyMap`) recovers the original spectral measure $\mu_D$: $$\text{cayleyMap}(\nu_D) = \mu_D$$

Let $\mathcal{H}$ be a Hilbert space and let $D$ be an object of bounded unitary spectral data, which includes a spectral measure $E_D$ on $\mathbb{C}$ supported on the unit circle excluding 1 ($\mathbb{T} \setminus \{1\}$). Let $E_{D, \mathbb{R}}$ be the associated real spectral measure on $\mathbb{R}$ constructed by pulling back $E_D$ along the Cayley transform $c: \mathbb{R} \to \mathbb{T} \setminus \{1\}$ defined by $c(x) = \frac{x - i}{x + i}$. For any spectral measure $\mu_S$ on $\mathbb{R}$, if the pushforward of $\mu_S$ along the Cayley transform (denoted as $\text{cayleyMap}(\mu_S)$) is equal to $E_D$, then $\mu_S$ must be equal to $E_{D, \mathbb{R}}$.