PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.CayleySpectralData.Basic

Given a continuous linear operator $U$ on a complex Hilbert space $H$, this definition represents the function mapping each spectral point $z \in \sigma(U)$ to its real part $\text{Re}(z)$. It is a continuous function from the spectrum $\sigma(U) \subseteq \mathbb{C}$ to the real numbers $\mathbb{R}$, and since the spectrum of a bounded operator is compact, this function is compactly supported.

For a continuous linear operator $U$ acting on a complex Hilbert space $H$, let $\sigma(U) \subseteq \mathbb{C}$ denote its spectrum. This definition represents the continuous function $f: \sigma(U) \to \mathbb{R}$ that maps each spectral point $z \in \sigma(U)$ to its imaginary part $\text{Im} z$. Since the spectrum of a bounded operator is compact, this function is an element of the space of compactly supported continuous maps on $\sigma(U)$.

Let $H$ be a complex Hilbert space and $U: H \to H$ be a normal bounded linear operator. For any vector $v \in H$, let $\mu_v$ be the scalar spectral measure associated with $U$ and $v$ via the continuous functional calculus. Then the integral of the identity function $z \mapsto z$ over the spectrum $\sigma(U)$ with respect to $\mu_v$ is equal to the sum of the integrals of its real and imaginary parts: $$\int_{\sigma(U)} z \, d\mu_v(z) = \int_{\sigma(U)} \operatorname{Re}(z) \, d\mu_v(z) + i \int_{\sigma(U)} \operatorname{Im}(z) \, d\mu_v(z)$$ where $\operatorname{Re}(z)$ and $\operatorname{Im}(z)$ denote the real and imaginary parts of the spectral values $z \in \sigma(U) \subseteq \mathbb{C}$.

Let $H$ be a complex Hilbert space and $U: H \to H$ be a normal bounded linear operator. For any vectors $x, y \in H$, let $\mu_{x,y}$ be the polarized spectral measure of $U$ obtained via the continuous functional calculus. Then the integral of the identity function $z \mapsto z$ over the spectrum of $U$ with respect to $\mu_{x,y}$ is equal to the inner product of $y$ and $Ux$: $$\int_{\sigma(U)} z \, d\mu_{x,y}(z) = \langle y, Ux \rangle$$ where $\sigma(U)$ denotes the spectrum of the operator $U$.

Given a self-adjoint operator $T$ acting on a Hilbert space $H$, this definition constructs its Cayley transform as a bounded continuous linear map from $H$ to $H$. The Cayley transform of a self-adjoint operator is a unitary operator, typically expressed in the form $(T - iI)(T + iI)^{-1}$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. For any vector $x \in H$, the value of the bounded operator representing the Cayley transform of $T$ applied to $x$ is equal to the value of the Cayley transform defined as a continuous linear map applied to $x$. That is, $\text{cayleyBoundedOperator}(T)x = \text{cayleyContinuousLinearMap}(T)x$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. Then the Cayley transform of $T$ is a normal operator, meaning it commutes with its adjoint.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. Then the Cayley transform of $T$ is a unitary operator in the space of bounded linear operators on $H$.

Given a self-adjoint operator $T$ acting on a complex Hilbert space $H$, let $U$ be its Cayley transform. This definition constructs the spectral measure of $U$ on the complex plane $\mathbb{C}$ by taking the spectral measure obtained from the continuous functional calculus (which is natively supported on the spectrum $\sigma(U) \subseteq \mathbb{C}$) and pushing it forward along the inclusion map $\sigma(U) \hookrightarrow \mathbb{C}$.

Let $H$ be a complex Hilbert space and $U$ be a normal bounded linear operator on $H$. Let $\mu_U$ be the spectral measure associated with $U$ via the continuous functional calculus, supported on the spectrum $\sigma(U) \subseteq \mathbb{C}$. For any vectors $x, y \in H$, the inner product $\langle y, U x \rangle$ can be reconstructed by integrating the identity function $z \mapsto z$ over the spectrum with respect to the complex scalar measure $\mu_{x, y}$ (where $\mu_{x, y}(E) = \langle y, \mu_U(E) x \rangle$): $$\int_{\sigma(U)} z \, d\langle y, \mu_U(z) x \rangle = \langle y, U x \rangle$$

Let $H$ be a complex Hilbert space and $U$ be a normal bounded linear operator on $H$. Let $\mu_U$ be the spectral measure associated with $U$ via the continuous functional calculus, which is initially defined on the spectrum $\sigma(U) \subseteq \mathbb{C}$. Let $\tilde{\mu}_U$ be the pushforward of this measure to the ambient complex plane $\mathbb{C}$ under the inclusion map $i: \sigma(U) \hookrightarrow \mathbb{C}$. For any vectors $x, y \in H$, the inner product $\langle y, U x \rangle$ can be reconstructed by integrating the identity function $z \mapsto z$ over $\mathbb{C}$ with respect to the complex scalar measure associated with $\tilde{\mu}_U$ and the vectors $x, y$: $$\int_{\mathbb{C}} z \, d\langle y, \tilde{\mu}_U(z) x \rangle = \langle y, U x \rangle$$

Let $H$ be a complex Hilbert space and $U: H \to H$ be a normal bounded linear operator. This definition constructs a "bounded normal spectral data" object for $U$ using its continuous functional calculus. The associated spectral measure $\mu$ is obtained by pushing forward the spectral measure from the spectrum $\sigma(U)$ to the complex plane $\mathbb{C}$ under the inclusion map $i: \sigma(U) \hookrightarrow \mathbb{C}$. This data object includes the reconstruction property that for all $x, y \in H$: $$\langle y, U x \rangle = \int_{\mathbb{C}} z \, d\langle y, \mu(z) x \rangle$$

Let $H$ be a complex Hilbert space. Given a unitary operator $U : H \to H$ (provided as a linear isometry) such that $1$ is not in the spectrum of $U$ (i.e., $1 \notin \sigma(U)$), this definition constructs the associated bounded unitary spectral data. This data includes a spectral measure $\mu$ on $\mathbb{C}$ that is supported on the unit circle excluding the point $1$ (the set $\{z \in \mathbb{C} \mid |z| = 1, z \neq 1\}$) and satisfies the reconstruction formula: $$\langle y, U x \rangle = \int_{\mathbb{C}} z \, d\langle y, \mu(z) x \rangle$$ for all vectors $x, y \in H$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. Let $U$ be the Cayley transform of $T$ (viewed as a bounded operator) and $E$ be its associated spectral measure on $\mathbb{C}$. For any vectors $x, y \in H$, the integral of the identity function $z \mapsto z$ with respect to the scalar measure $\langle y, E(\cdot) x \rangle$ is equal to the inner product of $y$ and $U x$: $$\int_{\mathbb{C}} z \, d\langle y, E x \rangle = \langle y, U x \rangle.$$

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. Let $E$ be the spectral measure on $\mathbb{C}$ associated with the Cayley transform of $T$. For any vectors $x, y \in H$, the scalar complex measure $\mu_{x, y}$ defined by $\mu_{x, y}(S) = \langle y, E(S)x \rangle$ for measurable sets $S \subseteq \mathbb{C}$ 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 $V$ be a bounded linear operator on $H$ that is reconstructed from $E$ in the sense that for all vectors $x, y \in H$, the identity function $z \mapsto z$ satisfies $$\int_{\mathbb{C}} z \, d\langle y, E x \rangle = \langle y, V x \rangle.$$ For any $a \in \mathbb{C}$ such that the singleton $\{a\}$ is measurable, and for any $x \in H$ such that the scalar complex measures $\langle y, E(\cdot) x \rangle$ have finite variation for all $y \in H$, the vector $E(\{a\})x$ is an eigenvector of $V$ with eigenvalue $a$. That is, $$V(E(\{a\})x) = a E(\{a\})x.$$

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

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. Let $U$ be the Cayley transform of $T$ (defined as the bounded operator $(T - iI)(T + iI)^{-1}$). 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 operator on $H$. Let $\mu$ be the spectral measure on the complex plane $\mathbb{C}$ associated with the Cayley transform of $T$. For any vectors $x, y \in H$, the identity function $f(z) = z$ is integrable with respect to the complex scalar measure $\mu_{x,y}$ defined by $\mu_{x,y}(S) = \langle y, \mu(S)x \rangle$ for measurable sets $S \subseteq \mathbb{C}$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. Let $\mu_U$ be the spectral measure on $\mathbb{C}$ associated with the Cayley transform of $T$. Then the measure of the singleton set $\{1\}$ is zero, i.e., $\mu_U(\{1\}) = 0$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator acting on $H$. Let $\mu$ be the spectral measure of the Cayley transform of $T$ (denoted formally as `cayleyBoundedSpectralMeasure T hT`). For any measurable set $S \subseteq \mathbb{C}$, it holds that $$\mu(S) = \mu(S \cap \{z \in \mathbb{C} \mid |z| = 1 \text{ and } z \neq 1\})$$ This identity expresses that the spectral measure of the Cayley transform is supported on the unit circle excluding the point $1$.

For any complex number $z \in \mathbb{C}$ on the unit circle (i.e., $|z| = 1$) such that $z \neq 1$, the inverse Cayley transform of $z$ satisfies the identity $$(\text{cayleyInverse } z) \cdot (1 - z) = i(1 + z)$$ where $i$ is the imaginary unit.

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

The function maps a complex number $z$ to $1 - z$ if $z$ is on the unit circle ($|z| = 1$), and to $0$ otherwise. This provides a globally bounded extension of the mapping $z \mapsto 1 - z$ from the unit circle to the complex plane.

The Cayley difference multiplier function $f: \mathbb{C} \to \mathbb{C}$, defined by $f(z) = 1 - z$ for complex numbers $z$ on the unit circle ($|z| = 1$) and $f(z) = 0$ otherwise, is a measurable function.

There exists a real constant $C$ such that for all complex numbers $z \in \mathbb{C}$, the norm of the Cayley difference multiplier satisfies $\|\text{cayleyDifferenceMultiplier}(z)\| \le C$. The Cayley difference multiplier is defined as the function mapping $z$ 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 operator on $H$. Let $U$ be the Cayley transform of $T$ (regarded as a bounded operator) and $\mu_U$ be the spectral measure of $U$ on the complex plane $\mathbb{C}$. Let $f(z)$ be the Cayley difference multiplier function, which is defined as $f(z) = 1 - z$ for $z$ on the unit circle ($|z| = 1$) and $f(z) = 0$ otherwise. Then the integral of $f$ with respect to the spectral measure $\mu_U$ in the weak operator topology is equal to the difference between the identity operator $I$ and the Cayley transform $U$: $$\int_{\mathbb{C}} f(z) \, d\mu_U(z) = I - U$$

Given a self-adjoint operator $T$ acting on a complex Hilbert space $H$, let $U$ be its Cayley transform. This definition constructs the bounded unitary spectral data for $U$, which consists of a spectral measure $\mu$ on $\mathbb{C}$ and the properties ensuring that $\mu$ is supported on the punctured unit circle $\{z \in \mathbb{C} : |z| = 1, z \neq 1\}$ and that $U$ can be reconstructed as the integral of the identity function with respect to this measure, i.e., $U = \int_{\mathbb{C}} z \, d\mu(z)$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint (partially defined) linear operator on $H$. This definition constructs the **real spectral measure** of $T$ as a spectral measure on the real line $\mathbb{R}$ with values in the operators on $H$. This measure is obtained by pulling back the spectral measure $\mu_U$ of the Cayley transform $U = (T - iI)(T + iI)^{-1}$ along the Cayley map $c: \mathbb{R} \to \mathbb{T} \setminus \{1\}$ defined by $c(x) = \frac{x - i}{x + i}$. Since $T$ is self-adjoint, the spectral measure of its unitary Cayley transform $U$ gives no mass to the point $\{1\}$, allowing the transport of the measure from the unit circle to the real line.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. Let $\mu_{\mathbb{R}}$ be the real spectral measure of $T$ and $\mu_U$ be the spectral measure of the Cayley transform of $T$ on the complex plane $\mathbb{C}$. If $c: \mathbb{R} \to \mathbb{C}$ is the Cayley map defined by $c(x) = \frac{x-i}{x+i}$, then the pushforward of $\mu_{\mathbb{R}}$ along $c$ is equal to $\mu_U$: $$c_* \mu_{\mathbb{R}} = \mu_U$$