PhyslibAlpha.AlgebraicFramework.HilbertSpace.Unbounded.CayleySpectralData.Construction

Given a continuous linear operator $U$ on a complex Hilbert space $H$, this function maps each point $z$ in the spectrum $\sigma(U)$ to its real part $\text{Re}(z)$. It is defined as a compactly supported continuous function from $\sigma(U)$ to $\mathbb{R}$.

Let $H$ be a complex Hilbert space and $U \in \mathcal{L}(H)$ be a continuous linear operator. This definition denotes the continuous, compactly supported function $f: \sigma(U) \to \mathbb{R}$ on the spectrum of $U$ that maps each spectral point $z \in \sigma(U)$ to its imaginary part $\text{Im}(z)$.

Let $H$ be a complex Hilbert space and $U$ be a normal bounded operator on $H$. For any vector $v \in H$, let $\mu_v$ denote the scalar spectral measure on the spectrum $\sigma(U)$ (the measure associated with $U$ and $v$ via the continuous functional calculus). Then the integral of the spectral points $z \in \sigma(U) \subseteq \mathbb{C}$ with respect to $\mu_v$ is the sum of the integral of their real parts and $i$ times the integral of their imaginary parts: $$\int_{\sigma(U)} z \, d\mu_v = \int_{\sigma(U)} \text{Re}(z) \, d\mu_v + i \int_{\sigma(U)} \text{Im}(z) \, d\mu_v$$

Let $H$ be a complex Hilbert space and $U: H \to H$ be a bounded normal operator. Let $\sigma(U) \subseteq \mathbb{C}$ denote its spectrum. For any vectors $x, y \in H$, let $\mu_{x, y}$ be the polarized complex scalar spectral measure associated with $U$. Then the integral of the identity function $f(z) = z$ over the spectrum with respect to this measure is equal to the inner product of $y$ and $Ux$: $$\int_{\sigma(U)} z \, d\mu_{x,y}(z) = \langle y, Ux \rangle$$

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. The Cayley transform of $T$ is the unitary operator $U = (T - iI)(T + iI)^{-1}$, where $i$ is the imaginary unit and $I$ is the identity operator. This definition expresses the Cayley transform of $T$ as a bounded continuous linear operator from $H$ to itself.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. For any vector $x \in H$, the Cayley transform of $T$, defined as $U = (T - iI)(T + iI)^{-1}$, satisfies the identity: $$Ux = (I - 2i(T + iI)^{-1})x$$ where $I$ is the identity operator and $i$ is the imaginary unit.

Let $H$ be a complex Hilbert space and let $T$ be a self-adjoint partial linear operator on $H$. The Cayley transform of $T$, denoted by $U = (T - iI)(T + iI)^{-1}$, is a normal operator (also referred to as star-normal), meaning it commutes with its adjoint: $UU^* = U^*U$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Then the Cayley transform of $T$ is a unitary operator. That is, if $U$ is the Cayley transform of $T$, then $U \in \text{unitary}(\mathcal{L}(H))$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $U = (T - iI)(T + iI)^{-1}$ be the bounded Cayley transform of $T$, which is a normal (unitary) operator. Let $E$ be the spectral measure (projection-valued measure) associated with $U$ via the continuous functional calculus, defined on the spectrum $\sigma(U) \subseteq \mathbb{C}$. This definition represents the spectral measure $\tilde{E}$ on the ambient complex plane $\mathbb{C}$ obtained by pushing forward $E$ along the inclusion map $\iota: \sigma(U) \hookrightarrow \mathbb{C}$.

Let $H$ be a complex Hilbert space and $U \in \mathcal{L}(H)$ be a bounded normal operator. Let $E$ be the spectral measure of $U$ (a projection-valued measure defined on the Borel $\sigma$-algebra of the spectrum $\sigma(U)$). For any vectors $x, y \in H$, the complex weak integral of the identity function $f(z) = z$ with respect to the scalar measure $\langle y, E(\cdot)x \rangle$ is equal to the inner product $\langle y, Ux \rangle$. That is, $$\int_{\sigma(U)} z \, d\langle y, E(z) x \rangle = \langle y, U x \rangle.$$

Let $H$ be a complex Hilbert space and $U \in \mathcal{L}(H)$ be a bounded normal operator. Let $E$ be the projection-valued spectral measure of $U$ defined on its spectrum $\sigma(U) \subseteq \mathbb{C}$, and let $\tilde{E}$ be the pushforward of $E$ to the ambient complex plane $\mathbb{C}$ via the inclusion map $\iota: \sigma(U) \hookrightarrow \mathbb{C}$. Then for any vectors $x, y \in H$, the integral of the identity function with respect to the scalar measure $\langle y, \tilde{E}(\cdot) x \rangle$ reconstructs the inner product of $U$: $$\int_{\mathbb{C}} z \, d\langle y, \tilde{E}(z) x \rangle = \langle y, U x \rangle$$

Let $H$ be a complex Hilbert space and $U \in \mathcal{L}(H)$ be a bounded normal operator. This definition constructs the **bounded normal spectral data** for $U$, which consists of: 1. A projection-valued measure (PVM) $E$ on the Borel $\sigma$-algebra of the complex plane $\mathbb{C}$. This measure is the pushforward of the spectral measure defined on the spectrum $\sigma(U) \subseteq \mathbb{C}$ via the inclusion map $\iota: \sigma(U) \hookrightarrow \mathbb{C}$. 2. A proof of the reconstruction property, which states that $U$ is the integral of the identity function with respect to this spectral measure in the weak operator sense: $$\int_{\mathbb{C}} z \, d\langle y, E(z) x \rangle = \langle y, U x \rangle$$ for all $x, y \in H$.

Let $H$ be a complex Hilbert space and $u : H \cong H$ be a unitary operator (represented as a linear isometric equivalence). Suppose that $1$ is not in the spectrum of $u$, denoted by $1 \notin \sigma(u)$. This definition constructs the **bounded unitary spectral data** for $u$, which consists of: 1. A projection-valued measure $E$ on the Borel $\sigma$-algebra of the complex plane $\mathbb{C}$. 2. A proof that the measure $E$ is supported on the punctured unit circle $\{z \in \mathbb{C} \mid |z| = 1, z \neq 1\}$. 3. A proof of the reconstruction property, which states that $u$ is the integral of the identity function with respect to $E$ in the weak operator sense: $$\int_{\mathbb{C}} z \, d\langle y, E(z) x \rangle = \langle y, u x \rangle$$ for all vectors $x, y \in H$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $U = (T - iI)(T + iI)^{-1}$ be the bounded Cayley transform of $T$, and let $E$ be the spectral measure on the complex plane $\mathbb{C}$ associated with $U$. For any vectors $x, y \in H$, the complex weak integral of the identity function with respect to $E$ satisfies the reconstruction property: $$\int_{\mathbb{C}} z \, d\langle y, E(z) x \rangle = \langle y, U x \rangle$$

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $E$ be the ambient spectral measure of the Cayley transform of $T$ (the projection-valued measure on $\mathbb{C}$ associated with the unitary operator $U = (T - iI)(T + iI)^{-1}$). For any vectors $x, y \in H$, the complex scalar measure $\mu_{x,y}$ defined by $\mu_{x,y}(S) = \langle y, E(S) x \rangle$ has finite total variation.

Let $H$ be a complex Hilbert space and $E$ be a spectral measure on $\mathbb{C}$ in the weak operator topology. Let $V: H \to H$ be a bounded linear operator that is reconstructed from the spectral measure $E$ via the identity $$\int_{\mathbb{C}} z \, d\langle y, E(z) x \rangle = \langle y, V x \rangle$$ for all $x, y \in H$. Suppose $a \in \mathbb{C}$ is a point such that the singleton set $\{a\}$ is measurable. For any vector $x \in H$ such that the complex scalar measures $\mu_{x,y}(S) = \langle y, E(S) x \rangle$ have finite total variation for all $y \in H$, the vector obtained by applying the spectral projection $E(\{a\})$ to $x$ is an eigenvector of $V$ with eigenvalue $a$: $$V(E(\{a\}) x) = a E(\{a\}) x$$

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $E$ be the spectral measure on $\mathbb{C}$ associated with the bounded Cayley transform of $T$. For any measurable set $S \subseteq \mathbb{C}$, the spectral measure $E(S)$ satisfies $$E(S) = E(S \cap \{z \in \mathbb{C} \mid |z| = 1\}).$$ This reflects the fact that the spectral measure of the Cayley transform (which is a unitary operator) is supported on the unit circle in the complex plane.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $U$ be the Cayley transform of $T$ (the bounded linear operator defined as $U = (T - iI)(T + iI)^{-1}$). For any vector $x \in H$, if $Ux = x$, then $x = 0$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $E$ be the spectral measure on $\mathbb{C}$ associated with the bounded Cayley transform of $T$, and for any two vectors $x, y \in H$, let $\mu_{x, y}$ be the complex scalar measure defined by $\mu_{x, y}(S) = \langle y, E(S)x \rangle$ for every measurable set $S \subseteq \mathbb{C}$. Then the identity function $f(z) = z$ is integrable with respect to the measure $\mu_{x, y}$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $E$ be the spectral measure (projection-valued measure) on $\mathbb{C}$ associated with the bounded Cayley transform $U = (T - iI)(T + iI)^{-1}$ of $T$. Then the spectral measure of the singleton set $\{1\}$ is zero, i.e., $$E(\{1\}) = 0$$

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. Let $E_U$ be the spectral measure on the complex plane $\mathbb{C}$ associated with the Cayley transform $U = (T - iI)(T + iI)^{-1}$ of $T$. For any measurable set $S \subseteq \mathbb{C}$, the spectral measure $E_U(S)$ satisfies: $$E_U(S) = E_U(S \cap \{z \in \mathbb{C} \mid \|z\| = 1, z \neq 1\})$$ This identity reflects that the spectral measure is supported on the unit circle and has no spectral mass at the point $1$.

For any complex number $z$ on the unit circle (i.e., $\|z\| = 1$) such that $z \neq 1$, the inverse scalar Cayley transform $c(z)$ satisfies the following identity in $\mathbb{C}$: $$c(z) \cdot (1 - z) = i(1 + z)$$ where $i$ denotes the imaginary unit and $c(z)$ is the real-valued function defined as $-\frac{\text{Im } z}{1 - \text{Re } z}$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$ with domain $D(T)$. Let $U$ denote the Cayley transform of $T$ (the bounded operator $(T - iI)(T + iI)^{-1}$). For any vector $x \in D(T)$, there exists a vector $v \in H$ such that $$x = v - Uv$$ and $$Tx = i(v + Uv),$$ where $i$ is the imaginary unit.

The function maps a complex number $z$ to $1 - z$ if it lies on the unit circle (i.e., $|z| = 1$), and to $0$ otherwise. This provides a globally bounded extension of the factor $1 - z$ from the unit circle to the entire complex plane.

The Cayley difference multiplier function, which maps a complex number $z \in \mathbb{C}$ to $1 - z$ if $|z| = 1$ and to $0$ otherwise, is a measurable function.

There exists a real constant $C$ such that for all complex numbers $z$, the norm of the Cayley difference multiplier, $\| \text{cayleyDifferenceMultiplier}(z) \|$, is at most $C$. Here, the Cayley difference multiplier is defined as the function mapping $z \in \mathbb{C}$ to $1 - z$ if $z$ lies on the unit circle ($\|z\| = 1$) and to $0$ otherwise.

For any complex number $z \in \mathbb{C}$ such that $\|z\| = 1$, the Cayley difference multiplier evaluated at $z$ is equal to $1 - z$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $U = (T - iI)(T + iI)^{-1}$ be the bounded Cayley transform of $T$. Let $\tilde{E}$ be the spectral measure on $\mathbb{C}$ associated with $U$ (the pushforward of the spectral measure on the spectrum $\sigma(U)$ to the complex plane). Let $f: \mathbb{C} \to \mathbb{C}$ be the Cayley difference multiplier, which is defined as $f(z) = 1 - z$ for $z$ on the unit circle ($|z|=1$) and $f(z) = 0$ otherwise. Then the spectral integral of $f$ with respect to $\tilde{E}$ satisfies: $$\int_{\mathbb{C}} f(z) \, d\tilde{E}(z) = I - U$$ where $I$ is the identity operator on $H$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $U = (T - iI)(T + iI)^{-1}$ be the unitary Cayley transform of $T$. This definition constructs a certificate of bounded unitary spectral data for $U$, which consists of: 1. The spectral measure $E_U$ on the complex plane $\mathbb{C}$ associated with $U$, obtained by pushing forward the spectral measure from the spectrum $\sigma(U)$ to the ambient field. 2. A proof that the spectral measure is supported on the unit circle and carries no mass at $1$, i.e., $E_U(S) = E_U(S \cap \{z \in \mathbb{C} \mid |z| = 1, z \neq 1\})$ for any measurable set $S$. 3. A proof of the reconstruction property, which identifies the operator $U$ with the spectral integral of the identity function: $U = \int_{\mathbb{C}} z \, dE_U(z)$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. The **real spectral measure** of $T$ is the spectral measure on the real line $\mathbb{R}$ obtained by taking the spectral measure of its Cayley transform $U = (T - iI)(T + iI)^{-1}$ (which is supported on the unit circle in $\mathbb{C} \setminus \{1\}$) and pulling it back to $\mathbb{R}$ via the inverse Cayley map.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Let $\mu_T$ be the **real spectral measure** of $T$ defined on the real line $\mathbb{R}$, and let $E_U$ be the **ambient spectral measure** on $\mathbb{C}$ associated with the bounded Cayley transform $U = (T - iI)(T + iI)^{-1}$. Let $\Phi : \mathbb{R} \to \mathbb{C}$ be the Cayley map defined by $\Phi(x) = \frac{x - i}{x + i}$. It holds that the pushforward of $\mu_T$ along $\Phi$ (denoted as $\Phi_* \mu_T$) is equal to $E_U$.