PhyslibAlpha.ProbabilisticTheory.HilbertSpace.TraceClass.Polar

For a bounded linear operator $T : H \to H$ on a complex Hilbert space $H$, the absolute value $|T|$ is a bounded linear operator on $H$ that satisfies the property $\|T x\| = \||T| x\|$ for all $x \in H$. It is the positive operator used in the polar decomposition $T = U|T|$.

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

For a bounded linear operator $T : H \to H$ on a complex Hilbert space $H$, let $|T|$ be its absolute value operator and $\overline{\text{ran}(|T|)}$ be the topological closure of the range of $|T|$. This definition is the linear map $|T| : H \to \overline{\text{ran}(|T|)}$ obtained by restricting the codomain of the absolute value operator to its closed range.

For a bounded linear operator $T$ on a complex Hilbert space $H$, let $|T|$ be its absolute value operator. This definition is the continuous linear inclusion map from the range of the absolute value operator, $\text{ran}(|T|)$, into the topological closure of that range, $\overline{\text{ran}(|T|)}$.

Let $H$ be a complex Hilbert space and $T: H \to H$ be a bounded linear operator. Let $|T|$ denote the absolute value operator of $T$, and let $\iota: \text{ran}(|T|) \to \overline{\text{ran}(|T|)}$ be the continuous linear inclusion map from the range of $|T|$ into its topological closure. For any element $x$ in the range of $|T|$, the value of the inclusion map applied to $x$ is equal to $x$ when viewed as an element of $H$.

For a bounded linear operator $T$ on a complex Hilbert space $H$, let $|T|$ be its absolute value operator and $\overline{\text{ran}(|T|)}$ be the topological closure of its range. The inclusion map from the range $\text{ran}(|T|)$ into $\overline{\text{ran}(|T|)}$ has a dense range in $\overline{\text{ran}(|T|)}$.

For a bounded linear operator $T : H \to H$ on a complex Hilbert space $H$, let $|T|$ be its absolute value operator and $\overline{\text{ran}(|T|)}$ be the topological closure of the range of $|T|$. Let $\text{absIntoRange}(T) : H \to \overline{\text{ran}(|T|)}$ be the map $|T|$ with its codomain restricted to this closed range, and let $\text{rangeInClosure}(T) : \text{ran}(|T|) \to \overline{\text{ran}(|T|)}$ be the natural inclusion map. For any $x \in H$, the identity $\text{absIntoRange}(T) x = \text{rangeInClosure}(T) (|T|x)$ holds, where $|T|x$ is viewed as an element of the range $\text{ran}(|T|)$.

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

Let $T : H \to H$ be a bounded linear operator on a complex Hilbert space $H$. Let $|T|$ denote the absolute value operator of $T$ and let $|T|_{\text{res}} : H \to \overline{\text{ran}(|T|)}$ be the version of $|T|$ where the codomain is restricted to the topological closure of its range. For any vector $x \in H$, the value $|T|_{\text{res}} x$ is equal to $|T| x$ when the former is viewed as an element of $H$.

For any bounded linear operator $T: H \to H$ on a complex Hilbert space $H$ and any vector $x \in H$, the norm of $T$ applied to $x$ is equal to the norm of its absolute value $|T|$ applied to $x$, such that $\|Tx\| = \||T|x\|$.

Let $T : H \to H$ be a bounded linear operator on a complex Hilbert space $H$. Let $|T|$ denote its absolute value operator and $\overline{\text{ran}(|T|)}$ denote the topological closure of the range of $|T|$. Let $|T|_{H \to \overline{\text{ran}(|T|)}}$ be the linear map obtained by restricting the codomain of $|T|$ to its closed range. For any vector $x \in H$, the norm of the image of $x$ under this restricted map is equal to the norm of the image of $x$ under the absolute value operator in $H$: $$\| |T|_{H \to \overline{\text{ran}(|T|)}} x \| = \| |T| x \|$$

For any bounded linear operator $T: H \to H$ on a complex Hilbert space $H$, the topological closure of the range of its absolute value operator $|T|$, denoted as $\overline{\text{ran}(|T|)}$, is a complete space.

For a bounded linear operator $T: H \to H$ on a complex Hilbert space $H$, let $|T|$ denote its absolute value operator and $\overline{\text{ran}(|T|)}$ denote the topological closure of the range of $|T|$. The operator `polarOnRange T` is the continuous linear map $U : \overline{\text{ran}(|T|)} \to H$ that satisfies $U(|T|x) = Tx$ for all $x \in H$. It represents the restriction of the polar factor to the closure of the range of $|T|$.

Let $H$ be a complex Hilbert space and $T: H \to H$ be a bounded linear operator. Let $|T|$ denote the absolute value operator of $T$, and let $U: \overline{\text{ran}(|T|)} \to H$ be the restricted polar factor (represented by `polarOnRange T`). For any vector $x \in H$, applying the polar factor $U$ to the image of $x$ under $|T|$ yields the image of $x$ under $T$, that is, $$U(|T|x) = Tx.$$

Let $H$ be a complex Hilbert space and $T : H \to H$ be a bounded linear operator. Let $|T|$ denote the absolute value of $T$, and let $\overline{\text{ran}(|T|)}$ denote the topological closure of the range of $|T|$. If $U : \overline{\text{ran}(|T|)} \to H$ is the continuous linear operator (the polar factor restricted to the range) defined by $U(|T|x) = Tx$ for all $x \in H$, then for every $x \in \overline{\text{ran}(|T|)}$, it holds that $$\|U x\| \leq \|x\|.$$

For a bounded linear operator $T: H \to H$ on a complex Hilbert space $H$, let $|T|$ be its absolute value operator. The polar factor $U: H \to H$ is the bounded linear operator obtained by extending the isometry $U(|T|x) = Tx$ from the closure of the range of $|T|$ (denoted $\overline{\text{ran}(|T|)}$) to the whole space $H$ by setting it to zero on the orthogonal complement $\left(\overline{\text{ran}(|T|)}\right)^\perp$. This operator $U$ is a partial isometry that satisfies the relation $T = U|T|$.

Let $H$ be a complex Hilbert space and $T: H \to H$ be a bounded linear operator. Let $|T|$ denote the absolute value operator of $T$ and $U$ denote the polar factor (the partial isometry) of $T$. For any vector $x \in H$, the following identity holds: $$U(|T|x) = Tx$$

Let $H$ be a complex Hilbert space and $T: H \to H$ be a bounded linear operator. Let $U$ be the polar factor (the partial isometry) of $T$. For any vector $x \in H$, the norm of $U x$ is less than or equal to the norm of $x$, i.e., $\|U x\| \leq \|x\|$.

Let $H$ be a complex Hilbert space and $T : H \to H$ be a bounded linear operator. Let $U$ be the polar factor of $T$ (the partial isometry defined such that $T = U|T|$). Then the operator norm of $U$ is less than or equal to $1$, i.e., $$\|U\| \le 1$$

Let $H$ be a complex Hilbert space and $T: H \to H$ be a bounded linear operator. Let $|T|$ denote the absolute value operator of $T$ and $U$ denote the polar factor (the partial isometry) of $T$. Then the operator $T$ can be decomposed as $$T = U |T|$$

Let $H$ be a complex Hilbert space and $T: H \to H$ be a bounded linear operator. Let $\overline{\text{ran}(|T|)}$ denote the topological closure of the range of the absolute value operator $|T|$, and let $U : \overline{\text{ran}(|T|)} \to H$ be the restricted polar factor of $T$. For any $y \in \overline{\text{ran}(|T|)}$, the norm of $U y$ is equal to the norm of $y$, i.e., $$\|U y\| = \|y\|$$

Let $H$ be a complex Hilbert space and $T: H \to H$ be a bounded linear operator. Let $|T|$ be the absolute value operator of $T$, and $\overline{\text{ran}(|T|)}$ be the topological closure of the range of $|T|$. Let $U: \overline{\text{ran}(|T|)} \to H$ be the isometry (denoted by `polarOnRange T`) that maps $|T|x$ to $Tx$. Then the adjoint $U^*$ is a left inverse of $U$, satisfying $U^* \circ U = I$, where $I$ is the identity operator on $\overline{\text{ran}(|T|)}$.

Let $H$ be a complex Hilbert space and $T: H \to H$ be a bounded linear operator. Let $U$ be the polar factor of $T$ (the partial isometry in the decomposition $T = U|T|$) and let $|T|$ be the absolute value operator of $T$. Then the composition of the adjoint of the polar factor $U^*$ with the polar factor $U$ is the orthogonal projection onto the closure of the range of $|T|$, denoted as $\overline{\text{ran}(|T|)}$. That is, $$U^* U = P_{\overline{\text{ran}(|T|)}}$$

Let $H$ be a complex Hilbert space and $T: H \to H$ be a bounded linear operator. Let $|T|$ denote the absolute value operator of $T$, and let $U$ be the polar factor (the partial isometry) of $T$ defined by the polar decomposition $T = U|T|$. Then it holds that $U^* T = |T|$, where $U^*$ is the adjoint of $U$.