PhyslibAlpha.AlgebraicFramework.HilbertSpace.TraceClass.Polar

For a bounded linear operator $T \in \mathcal{L}(H)$ acting on a complex Hilbert space $H$, the function returns the absolute value of the operator $|T|$, which is defined by applying the absolute value function to $T$ via the continuous functional calculus.

For a bounded linear operator $T \in \mathcal{L}(H)$ on a complex Hilbert space $H$, let $|T|$ denote its absolute value. This definition represents the topological closure of the range of $|T|$, which is the closed subspace of $H$ denoted by $\overline{\text{ran}(|T|)}$.

For a bounded linear operator $T \in \mathcal{L}(H)$ acting on a complex Hilbert space $H$, let $|T|$ denote its absolute value. The map `absIntoRange` is the linear map $|T| : H \to \overline{\text{ran}(|T|)}$ defined by restricting the codomain of the absolute value operator to the closure of its range.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$, $\text{ran}(|T|)$ denote its range, and $\overline{\text{ran}(|T|)}$ denote the topological closure of that range. This definition is the canonical continuous linear inclusion map from $\text{ran}(|T|)$ into $\overline{\text{ran}(|T|)}$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$, $\text{ran}(|T|)$ denote its range, and $\overline{\text{ran}(|T|)}$ denote the topological closure of that range. Let $\iota: \text{ran}(|T|) \to \overline{\text{ran}(|T|)}$ be the canonical continuous linear inclusion map (denoted as `rangeInClosure T`). For any vector $x \in \text{ran}(|T|)$, it holds that $\iota(x) = x$ when the result is viewed as an element of the Hilbert space $H$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$, $\text{ran}(|T|)$ denote its range, and $\overline{\text{ran}(|T|)}$ denote the topological closure of that range. The canonical inclusion map $\iota : \text{ran}(|T|) \to \overline{\text{ran}(|T|)}$ has a dense range.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$, $\text{ran}(|T|)$ its range, and $\overline{\text{ran}(|T|)}$ the closure of its range. Let $|T|_{\text{cl}} : H \to \overline{\text{ran}(|T|)}$ be the operator $|T|$ with its codomain restricted to its closed range, and let $\iota : \text{ran}(|T|) \to \overline{\text{ran}(|T|)}$ be the canonical inclusion map. For any vector $x \in H$, it holds that $$|T|_{\text{cl}} x = \iota(|T|x)$$ where $|T|x$ in the right-hand side is treated as an element of the subspace $\text{ran}(|T|)$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$, and let $\overline{\text{ran}(|T|)}$ denote the closure of its range. The range of the linear map $|T| : H \to \overline{\text{ran}(|T|)}$ is dense in $\overline{\text{ran}(|T|)}$.

For a bounded linear operator $T \in \mathcal{L}(H)$ acting on a complex Hilbert space $H$, let $|T|$ denote its absolute value. Let $|T|_{\to \overline{\text{ran}(|T|)}}$ be the linear map from $H$ to the closure of the range of $|T|$ (denoted as `absIntoRange T`). For any vector $x \in H$, applying this map to $x$ and viewing the result as an element of $H$ is equal to $|T|x$.

Let $H$ be a complex Hilbert space. For any bounded linear operator $T \in \mathcal{L}(H)$ and any vector $x \in H$, the norm of $T x$ is equal to the norm of $|T| x$, where $|T|$ denotes the absolute value of the operator $T$. That is, $\|T x\| = \||T| x\|$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$. For any vector $x \in H$, the norm of $|T|x$ viewed as an element of the closed range $\overline{\text{ran}(|T|)}$ is equal to its norm viewed as an element of $H$. That is, $\||T|x\| = \||T|x\|$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$. The topological closure of the range of $|T|$, denoted by $\overline{\text{ran}(|T|)}$, is a complete space.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$ and $\overline{\text{ran}(|T|)}$ denote the topological closure of the range of $|T|$. The map `polarOnRange T` is the continuous linear map $U : \overline{\text{ran}(|T|)} \to H$ obtained by extending the assignment $U(|T|x) = Tx$ from the range of $|T|$ to its closure. This extension is well-defined because $\|Tx\| = \||T|x\|$ for all $x \in H$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$, and let $U: \overline{\text{ran}(|T|)} \to H$ be the partial isometry (denoted as `polarOnRange T`) obtained by extending the assignment $U(|T|x) = Tx$ to the closure of the range of $|T|$. Then for any $x \in H$, it holds that $$U(|T|x) = Tx$$ where $|T|x$ is viewed as an element of the closed subspace $\overline{\text{ran}(|T|)}$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$ and $\overline{\text{ran}(|T|)}$ be the closure of its range. Let $U : \overline{\text{ran}(|T|)} \to H$ be the continuous linear map (denoted by `polarOnRange T`) obtained by extending the assignment $U(|T|v) = Tv$ for $v \in H$. Then for any $x \in \overline{\text{ran}(|T|)}$, the norm of $U x$ is bounded by the norm of $x$: $$\|U x\| \le \|x\|$$

For a bounded linear operator $T \in \mathcal{L}(H)$ on a complex Hilbert space $H$, the `polarFactor` is the bounded linear operator $U : H \to H$ defined as the composition $U = V \circ P$, where $P: H \to \overline{\text{ran}(|T|)}$ is the orthogonal projection onto the closure of the range of the absolute value $|T|$, and $V: \overline{\text{ran}(|T|)} \to H$ is the partial isometry (`polarOnRange T`) that maps $|T|x$ to $Tx$ for all $x \in H$. This construction results in a partial isometry $U$ that satisfies $T = U|T|$ and acts as zero on the orthogonal complement $(\text{ran}(|T|))^\perp$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$ and $U$ (the `polarFactor` of $T$) be the partial isometry associated with $T$ in its polar decomposition. Then for any vector $x \in H$, the operator $U$ maps $|T|x$ to $Tx$: $$U(|T|x) = Tx$$

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $U$ (denoted as `polarFactor T`) be the partial isometry obtained from the polar decomposition of $T$. For any vector $x \in H$, the norm of the image of $x$ under $U$ is bounded by the norm of $x$: $$\|U x\| \leq \|x\|$$

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $U$ (denoted as `polarFactor T`) be the partial isometry in the polar decomposition of $T$, such that $T = U|T|$. Then the operator norm of $U$ satisfies $\|U\| \le 1$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$ and $U$ denote the partial isometry given by the polar factor of $T$. Then the composition of $U$ and $|T|$ satisfies $U |T| = T$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$ and $\overline{\text{ran}(|T|)}$ denote the topological closure of the range of $|T|$. Let $U: \overline{\text{ran}(|T|)} \to H$ be the continuous linear map (denoted as `polarOnRange T`) obtained by extending the assignment $U(|T|x) = Tx$ to the closure of the range. For any vector $y \in \overline{\text{ran}(|T|)}$, it holds that $\|Uy\| = \|y\|$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$, and let $\overline{\text{ran}(|T|)}$ be the topological closure of its range. Let $U: \overline{\text{ran}(|T|)} \to H$ be the continuous linear map `polarOnRange T` (the isometry that extends $U(|T|x) = Tx$). Then $U^* \circ U = I$, where $U^*$ is the adjoint of $U$ and $I$ is the identity operator on $\overline{\text{ran}(|T|)}$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $U$ be the partial isometry (the `polarFactor`) such that $T = U|T|$, where $|T|$ is the absolute value of $T$. Then the operator $U^* U$ is equal to the orthogonal projection onto the topological closure of the range of $|T|$, denoted by $\overline{\text{ran}(|T|)}$.

Let $H$ be a complex Hilbert space and $T \in \mathcal{L}(H)$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$ and $U$ denote the polar factor of $T$ (the partial isometry provided by the polar decomposition). Then the product of the adjoint of the polar factor and the operator $T$ recovers the absolute value, i.e., $U^* T = |T|$.