PhyslibAlpha.AlgebraicFramework.HilbertSpace.Unbounded.Cayley.Measure

Given a weak-operator-topology (WOT) spectral measure $\mu$ on the real line $\mathbb{R}$ taking values in the bounded operators on a Hilbert space $H$, the Cayley map of $\mu$ is the pushforward spectral measure on the complex plane $\mathbb{C}$ along the Cayley transform $\Phi : \mathbb{R} \to \mathbb{C}$. For any measurable set $E \subseteq \mathbb{C}$, the resulting measure is defined by $(\Phi_* \mu)(E) = \mu(\Phi^{-1}(E))$.

Let $\nu$ be a weak-operator-topology (WOT) spectral measure on the complex plane $\mathbb{C}$ with values in the bounded operators on a Hilbert space $H$. This definition constructs a WOT spectral measure on the real line $\mathbb{R}$ by pushing $\nu$ forward along the inverse Cayley transform $g: \mathbb{C} \to \mathbb{R}$. For any measurable set $S \subseteq \mathbb{R}$, the resulting measure is given by $\nu(g^{-1}(S))$.

Let $H$ be a Hilbert space and $\mu$ be a weak-operator-topology (WOT) spectral measure on the real line $\mathbb{R}$ taking values in the bounded operators on $H$. Let $\Phi: \mathbb{R} \to \mathbb{C}$ be the Cayley transform and $g: \mathbb{C} \to \mathbb{R}$ be the inverse Cayley transform. The pushforward of $\mu$ along $\Phi$ (the Cayley map) followed by the pushforward along $g$ (the inverse Cayley map) returns the original measure $\mu$: $$\text{cayleyInverseMap} (\text{cayleyMap } \mu) = \mu$$

Let $H$ be a Hilbert space and let $\mu$ and $\nu$ be weak-operator-topology (WOT) spectral measures on the real line $\mathbb{R}$ taking values in the bounded operators on $H$. If the pushforwards of $\mu$ and $\nu$ under the Cayley transform $\Phi: \mathbb{R} \to \mathbb{C}$ (denoted by the `cayleyMap`) are equal, then $\mu = \nu$.

A spectral measure $\nu$ on the complex plane $\mathbb{C}$ is said to be Cayley-supported if it is concentrated on the unit circle excluding the point $1$. Formally, for every measurable set $S \subseteq \mathbb{C}$, the measure satisfies $\nu(S) = \nu(S \cap \{z \in \mathbb{C} \mid |z| = 1, z \neq 1\})$. This condition ensures that the measure is supported on the image of the real line under the Cayley transform, avoiding the point at infinity (represented by $z=1$).

Let $H$ be a Hilbert space and let $\nu$ be a weak-operator-topology (WOT) spectral measure on the complex plane $\mathbb{C}$ taking values in the bounded operators on $H$. If $\nu$ is Cayley-supported—meaning that for every measurable set $S \subseteq \mathbb{C}$, $\nu(S) = \nu(S \cap \{z \in \mathbb{C} \mid |z| = 1, z \neq 1\})$—then applying the pushforward of the inverse Cayley transform followed by the pushforward of the Cayley transform to $\nu$ recovers the original measure. Mathematically, this states that $\text{cayleyMap}(\text{cayleyInverseMap}(\nu)) = \nu$ for any such supported measure $\nu$.

Let $H$ be a Hilbert space and $\mu$ be a weak-operator-topology (WOT) spectral measure on the real line $\mathbb{R}$ with values in the bounded operators on $H$. The pushforward spectral measure $\Phi_* \mu$ on the complex plane $\mathbb{C}$ along the Cayley transform $\Phi : \mathbb{R} \to \mathbb{C}$ is Cayley-supported. That is, $\Phi_* \mu$ is concentrated on the unit circle excluding the point $1$, satisfying $(\Phi_* \mu)(S) = (\Phi_* \mu)(S \cap \{z \in \mathbb{C} \mid |z| = 1, z \neq 1\})$ for any measurable set $S \subseteq \mathbb{C}$.

Let $H$ be a Hilbert space. This definition establishes a bijective equivalence (isomorphism) between the set of weak-operator-topology (WOT) spectral measures on the real line $\mathbb{R}$ and the set of WOT spectral measures on the complex plane $\mathbb{C}$ that are Cayley-supported: $$\text{WOTSpectralMeasure}(\mathbb{R}, H) \simeq \{ \nu : \text{WOTSpectralMeasure}(\mathbb{C}, H) \mid \text{CayleySupported}(\nu) \}$$ A spectral measure $\nu$ is Cayley-supported if it is concentrated on the unit circle excluding the point $1$, i.e., its support is contained in $\{z \in \mathbb{C} \mid |z| = 1, z \neq 1\}$. The forward map sends a real spectral measure $\mu$ to its pushforward under the Cayley transform $\Phi: \mathbb{R} \to \mathbb{C}$, and the inverse map sends a Cayley-supported measure $\nu$ to its pushforward under the inverse Cayley transform $g: \mathbb{C} \to \mathbb{R}$.

Let $H$ be a Hilbert space and $\mu$ be a weak-operator-topology (WOT) spectral measure on $\mathbb{R}$ with values in the bounded operators on $H$. Let $\Phi: \mathbb{R} \to \mathbb{C}$ denote the Cayley transform, and let $\Phi_* \mu$ be the pushforward spectral measure on $\mathbb{C}$ along $\Phi$. For any real-valued function $g: \mathbb{C} \to \mathbb{R}$ and any vectors $x, y \in H$, if $g$ is almost everywhere strongly measurable with respect to the pushforward of the variation of the scalar measure $\mu_{x,y}$ and $g \circ \Phi$ is integrable with respect to $\mu_{x,y}$, then $$\int_{\mathbb{C}} g(z) \, d\langle y, (\Phi_* \mu)(z) x \rangle = \int_{\mathbb{R}} g(\Phi(t)) \, d\langle y, \mu(t) x \rangle.$$ where $\langle y, \mu(\cdot) x \rangle$ is the complex-valued scalar measure mapping a measurable set $S \subseteq \mathbb{R}$ to the inner product $\langle y, \mu(S) x \rangle$.