PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.Cayley.Measure

Given a weak-operator spectral measure $\mu$ on the real numbers $\mathbb{R}$ with values in the bounded operators on a Hilbert space $H$, this definition constructs the pushforward spectral measure $c_* \mu$ on the complex plane $\mathbb{C}$ along the Cayley transform $c: \mathbb{R} \to \mathbb{C}$. The Cayley transform is defined as $c(x) = \frac{x - i}{x + i}$. The resulting measure $c_* \mu$ is defined such that for any measurable set $E \subseteq \mathbb{C}$, $(c_* \mu)(E) = \mu(c^{-1}(E))$.

Given a weak-operator spectral measure $\nu$ on the complex plane $\mathbb{C}$ taking values in the projections on a Hilbert space $H$, the function `cayleyInverseMap` defines a new weak-operator spectral measure on the real line $\mathbb{R}$. This is constructed by taking the pushforward of $\nu$ along the inverse Cayley transform $f: \mathbb{C} \to \mathbb{R}$. For $z \neq 1$, the transformation is given by $f(z) = \frac{-\text{Im} z}{1 - \text{Re} z}$. For any measurable set $B \subseteq \mathbb{R}$, the value of the resulting spectral measure is $\nu(f^{-1}(B))$.

Let $H$ be a Hilbert space and let $\mu$ be a weak-operator spectral measure on the real line $\mathbb{R}$ taking values in the bounded operators on $H$. Let $c: \mathbb{R} \to \mathbb{C}$ be the Cayley transform defined by $c(x) = \frac{x - i}{x + i}$, and let $f: \mathbb{C} \to \mathbb{R}$ be the inverse Cayley transform. If $c_* \mu$ denotes the pushforward of $\mu$ along $c$, and $f_*$ denotes the pushforward of a spectral measure along $f$, then $f_* (c_* \mu) = \mu$.

Let $H$ be a Hilbert space and let $\mu_1, \mu_2$ be weak-operator spectral measures on $\mathbb{R}$ with values in the bounded operators on $H$. Let $c_*$ denote the pushforward operation along the Cayley transform $c: \mathbb{R} \to \mathbb{C}$, defined by $c(x) = \frac{x - i}{x + i}$. If $c_* \mu_1 = c_* \mu_2$, then $\mu_1 = \mu_2$. In other words, the Cayley pushforward map is injective.

Let $\nu$ be a spectral measure on $\mathbb{C}$ acting on a Hilbert space $H$. The property **Cayley supported** holds for $\nu$ if the measure is supported on the unit circle excluding the point $1$. Specifically, for any measurable set $S \subseteq \mathbb{C}$, the spectral projection satisfies: $$\nu(S) = \nu(S \cap \{z \in \mathbb{C} \mid |z| = 1, z \neq 1\})$$

Let $H$ be a Hilbert space and $\nu$ be a weak-operator spectral measure on $\mathbb{C}$ with values in the bounded operators on $H$. If $\nu$ is **Cayley supported** (that is, $\nu$ is supported on the unit circle excluding the point $1$), then the pushforward of $\nu$ along the inverse Cayley transform $f: \mathbb{C} \to \mathbb{R}$ followed by the pushforward along the Cayley transform $c: \mathbb{R} \to \mathbb{C}$ recovers the original measure: $$c_*(f_* \nu) = \nu$$ where $c_*(f_* \nu)$ denotes `cayleyMap (cayleyInverseMap ν)`.

Let $H$ be a Hilbert space and $\mu$ be a weak-operator spectral measure on the real numbers $\mathbb{R}$ with values in the bounded operators on $H$. Let $c_* \mu$ be the pushforward of $\mu$ along the Cayley transform $c: \mathbb{R} \to \mathbb{C}$ defined by $c(x) = \frac{x - i}{x + i}$. Then the spectral measure $c_* \mu$ is **Cayley supported**, meaning it is supported on the unit circle excluding the point $1$. That is, for any measurable set $S \subseteq \mathbb{C}$, the spectral projection satisfies: $$(c_* \mu)(S) = (c_* \mu)(S \cap \{z \in \mathbb{C} \mid |z| = 1, z \neq 1\})$$

Let $H$ be a Hilbert space. There is an equivalence (a bijective correspondence) between the set of weak-operator spectral measures on the real line $\mathbb{R}$ and the set of weak-operator spectral measures on the complex plane $\mathbb{C}$ that are **Cayley supported**. A spectral measure $\nu$ on $\mathbb{C}$ is Cayley supported if its support is contained in the unit circle excluding the point $1$, i.e., for any measurable set $S \subseteq \mathbb{C}$, $\nu(S) = \nu(S \cap \{z \in \mathbb{C} \mid |z| = 1, z \neq 1\})$. This equivalence is established by the pushforward along the Cayley transform $c: \mathbb{R} \to \mathbb{C}$, defined by $c(x) = \frac{x - i}{x + i}$, and the pushforward along its inverse.

Let $\mu$ be a weak-operator spectral measure on $\mathbb{R}$ with values in the bounded operators on a Hilbert space $H$. Let $c: \mathbb{R} \to \mathbb{C}$ be the Cayley transform defined by $c(x) = \frac{x - i}{x + i}$, and let $c_* \mu$ be the pushforward spectral measure on $\mathbb{C}$ along $c$. For any function $g: \mathbb{C} \to \mathbb{R}$ and vectors $x, y \in H$, suppose that $g$ is almost everywhere strongly measurable with respect to the pushforward of the total variation of the scalar complex measure $\mu_{x,y}(E) = \langle y, \mu(E)x \rangle$, and that $g \circ c$ is integrable with respect to $\mu_{x,y}$. Then the weak integral of $g$ with respect to the pushforward measure $c_* \mu$ equals the weak integral of $g \circ c$ with respect to the original measure $\mu$: $$\int_{\mathbb{C}} g \, d\langle y, (c_* \mu)(\cdot) x \rangle = \int_{\mathbb{R}} (g \circ c) \, d\langle y, \mu(\cdot) x \rangle$$