PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.Cayley.Basic

The function maps a real number $x$ to the complex number $c(x) = \frac{x - i}{x + i}$, where $i$ is the imaginary unit. This transformation maps the real line $\mathbb{R}$ to the unit circle in the complex plane $\mathbb{C}$, excluding the point $1$.

For a complex number $z$, this function computes the inverse Cayley transform, which maps $z$ to a real number $x$. For $z \neq 1$, it is given by the formula $x = \frac{-\text{Im} z}{1 - \text{Re} z}$, which satisfies the relation $z = \frac{x - i}{x + i}$. At the point $z = 1$ (which corresponds to infinity), the function is assigned an arbitrary real value.

For any real number $x$, the scalar Cayley transform $c(x) = \frac{x - i}{x + i}$ is not equal to $1$.

For any real number $x$, the real part of its Cayley transform $c(x) = \frac{x - i}{x + i}$ is given by $$\operatorname{Re}(c(x)) = \frac{x^2 - 1}{x^2 + 1},$$ where $i$ is the imaginary unit.

For any real number $x$, the imaginary part of its scalar Cayley transform $c(x) = \frac{x - i}{x + i}$ is given by $\frac{-2x}{x^2 + 1}$.

For any real number $x$, the inverse Cayley transform of the Cayley transform of $x$ is equal to $x$. That is, if $c(x) = \frac{x - i}{x + i}$ is the Cayley transform mapping the real line to the unit circle, then $c^{-1}(c(x)) = x$.

For any real number $x$, the norm of the Cayley transform $c(x) = \frac{x - i}{x + i}$ is equal to $1$.

For any complex number $z \in \mathbb{C}$ such that $\|z\| = 1$ and $z \neq 1$, applying the Cayley transform to the inverse Cayley transform of $z$ returns $z$, i.e., $\text{cayley}(\text{cayleyInverse}(z)) = z$. The Cayley transform is defined as $\text{cayley}(x) = \frac{x - i}{x + i}$ for $x \in \mathbb{R}$, which maps the real line to the unit circle excluding $1$.

The scalar Cayley transform $\text{cayley} : \mathbb{R} \to \mathbb{C}$, defined by $\text{cayley}(x) = \frac{x - i}{x + i}$ where $i$ is the imaginary unit, is a measurable function.

The inverse Cayley transform $\text{cayleyInverse} : \mathbb{C} \to \mathbb{R}$ is a measurable function.

Let $H$ be a complex Hilbert space and $T$ be a partial linear map on $H$. The Cayley transform of $T$, denoted $\text{cayleyPMap}(T)$, is the partial linear map defined by the expression $$(T - i I) \circ (T + i I)^{-1}$$ where $i$ is the imaginary unit and $I$ is the identity operator. This construction represents the operator-theoretic version of the scalar Cayley transform $c(x) = \frac{x - i}{x + i}$ before it is restricted or identified as a bounded/unitary operator.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partial linear operator on $H$. Then the domain of the Cayley transform of $T$, defined as $\text{cayleyPMap}(T) = (T - i I)(T + i I)^{-1}$, is the entire space $H$.

Given a partial linear map $A$ on a complex Hilbert space $H$ whose domain is the entire space ($\text{dom}(A) = H$) and which is continuous, this function constructs the corresponding continuous (bounded) linear operator $A \in \mathcal{L}(H)$.

Let $H$ be a complex Hilbert space. Suppose $A$ is a partial linear map on $H$ whose domain is the entire space ($\text{dom}(A) = H$) and which is continuous. Then for any $x \in H$, the value of the continuous linear operator $\tilde{A} \in \mathcal{L}(H)$ (induced by $A$) applied to $x$ is equal to the value of the partial map $A$ applied to $x$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partially defined linear operator on $H$. Then the Cayley transform of $T$, denoted $\text{cayleyPMap}(T)$, satisfies the identity: $$\text{cayleyPMap}(T) = I - 2i(T + iI)^{-1}$$ where $I$ is the identity operator on $H$ and $i$ is the imaginary unit.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partially defined linear operator on $H$. The Cayley transform of $T$, denoted $\text{cayleyPMap}(T)$, satisfies the identity: $$\text{cayleyPMap}(T) = 1 - 2i(T - (-i)I)^{-1}$$ where $i$ is the imaginary unit and $1$ (or $I$) denotes the identity operator.

Given a self-adjoint partially defined linear operator $T$ on a complex Hilbert space $H$, this definition constructs the corresponding bounded (continuous) linear operator $U \in \mathcal{L}(H)$ via the Cayley transform: $$U = (T - iI)(T + iI)^{-1}$$ where $i$ is the imaginary unit and $I$ is the identity operator. The construction utilizes the fact that for a self-adjoint operator, the imaginary unit $-i$ (and $+i$) lies in the resolvent set, ensuring that $(T + iI)^{-1}$ is a bounded operator defined on the entire space $H$.

For a bounded linear operator $L: H \to H$ on a complex Hilbert space $H$, this function constructs the corresponding partially defined linear map (of type `LinearPMap`) whose domain is the entire space $H$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint (partially defined) linear operator on $H$. The Cayley transform of $T$, defined as the partial linear map $\text{cayleyPMap}(T) = (T - i I)(T + i I)^{-1}$, is equal to the representation of the bounded Cayley transform operator $\text{cayleyContinuousLinearMap}(T)$ as a partial linear map. Here $i$ denotes the imaginary unit and $I$ denotes the identity operator.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint operator on $H$. For any vector $y$ in the domain of $T$, the following equality holds: $$\|Ty - iy\| = \|Ty + iy\|$$ where $i$ is the imaginary unit and $\|\cdot\|$ denotes the norm on $H$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partially defined linear operator on $H$. Let $U$ be the Cayley transform of $T$, which is the continuous linear operator $U = (T - iI)(T + iI)^{-1}$, where $i$ is the imaginary unit and $I$ is the identity operator. For any vector $y$ in the domain of $T + iI$ and any $x \in H$, if $(T + iI)y = x$, then $U x = (T - iI)y$.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint (possibly partially defined) linear operator on $H$. Let $U$ be the Cayley transform of $T$, which is the bounded linear operator defined by $U = (T - iI)(T + iI)^{-1}$, where $i$ is the imaginary unit and $I$ is the identity operator. For any vector $x \in H$, the norm of the Cayley transform of $x$ is equal to the norm of $x$: $$\|U x\| = \|x\|$$

Let $H$ be a complex Hilbert space and let $T$ be a self-adjoint partially defined linear operator on $H$. Then the Cayley transform of $T$, defined as the bounded linear operator $U = (T - iI)(T + iI)^{-1}$, is an isometry on $H$.

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

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partially defined linear operator on $H$. The Cayley transform of $T$ is the unitary operator $U: H \to H$ (an isometric linear isomorphism) defined by $$U = (T - iI)(T + iI)^{-1}$$ where $i$ is the imaginary unit and $I$ is the identity operator. This construction promotes the bounded Cayley transform to a full isometric equivalence, reflecting that for self-adjoint operators, the Cayley transform is both an isometry and a surjection.

Let $H$ be a complex Hilbert space and $T$ be a self-adjoint partially defined linear operator on $H$. For any vector $x \in H$, the result of applying the unitary Cayley transform of $T$ to $x$ is equal to the result of applying the bounded (continuous) linear map version of the Cayley transform of $T$ to $x$. Both definitions represent the operator $U = (T - iI)(T + iI)^{-1}$, where $i$ is the imaginary unit and $I$ is the identity operator.