PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.Conjugation

Given Hilbert spaces $H$ and $H'$, a unitary operator $u : H \to H'$, and a bounded linear operator $A$ on $H$, this definition represents the conjugated operator $u A u^{-1}$ acting on $H'$. The resulting operator is considered as an element of the space of bounded operators equipped with the Weak Operator Topology (WOT).

Given Hilbert spaces $H$ and $H'$ and a unitary operator $u : H \cong H'$, this map is the additive homomorphism from the space of bounded linear operators on $H$ to those on $H'$, both equipped with the Weak Operator Topology, defined by the conjugation $A \mapsto u A u^{-1}$.

Let $H$ and $H'$ be complex Hilbert spaces, and let $u : H \cong H'$ be a unitary operator (isometric linear isomorphism). The map that sends a bounded linear operator $A$ on $H$ to its conjugation $u A u^{-1}$ on $H'$ is continuous with respect to the weak operator topology on both spaces.

Let $H$ and $H'$ be complex Hilbert spaces, and let $u : H \cong H'$ be a unitary operator. For any bounded linear operators $A$ and $B$ on $H$, the unitary conjugation of their product $AB$ by $u$ is equal to the product of the unitary conjugations of $A$ and $B$ by $u$. That is, $$\text{unitaryConj}_u(AB) = \text{unitaryConj}_u(A) \text{unitaryConj}_u(B)$$ where $\text{unitaryConj}_u(A)$ denotes the operator $u A u^{-1}$ acting on $H'$.

Given Hilbert spaces $H$ and $H'$, let $u : H \to H'$ be a unitary operator (isometric linear isomorphism). For any bounded linear operator $A$ on $H$, the conjugation of the adjoint operator $A^*$ by $u$ is equal to the adjoint of the conjugated operator $A$: $$\text{unitaryConj}_u(A^*) = (\text{unitaryConj}_u(A))^*$$ where $A^*$ (or $\text{star } A$) denotes the adjoint of the operator $A$ and $\text{unitaryConj}_u(A)$ denotes the operator $u A u^{-1}$ acting on $H'$.

Let $H$ and $H'$ be complex Hilbert spaces and $u : H \to H'$ be a unitary operator (isometric linear equivalence). The unitary conjugation of the identity operator $I_H$ on $H$ by $u$ is the identity operator $I_{H'}$ on $H'$, i.e., $u I_H u^{-1} = I_{H'}$.

Let $H$ and $H'$ be complex Hilbert spaces, and let $u : H \cong H'$ be a unitary operator (an isometric linear isomorphism). For a spectral measure $\mu$ on $H$ defined with respect to the weak operator topology, this definition constructs a transported spectral measure on $H'$. For any measurable set $S \subseteq \alpha$, the value of the new spectral measure is given by the conjugation of the projection $\mu(S)$ by $u$: $$(u \mu u^{-1})(S) = u \mu(S) u^{-1}$$ where $u^{-1}$ is the adjoint (or inverse) of the unitary operator $u$.

Let $H$ and $H'$ be complex Hilbert spaces, and let $u : H \cong H'$ be a unitary operator (an isometric linear isomorphism). For a spectral measure $\mu$ on $H$ defined with respect to the weak operator topology and any measurable set $S \subseteq \alpha$, the value of the unitarily conjugated spectral measure $(u \mu u^{-1})$ at $S$ is equal to the unitary conjugation of the operator $\mu(S)$ by $u$. That is: $$(u \mu u^{-1})(S) = u \mu(S) u^{-1}$$

Let $H$ and $H'$ be complex Hilbert spaces, and let $u : H \to H'$ be a unitary operator (an isometric linear isomorphism). Let $\mu$ be a spectral measure on $H$ defined with respect to the weak operator topology. For any vectors $x, y \in H'$ and any measurable set $S$, the scalar measure associated with the unitarily conjugated spectral measure $u \mu u^{-1}$ satisfies: $$\langle y, (u \mu(S) u^{-1}) x \rangle = \langle u^{-1} y, \mu(S) u^{-1} x \rangle$$ where $u^{-1}$ denotes the inverse (and adjoint) of the unitary operator $u$.

Let $H$ and $H'$ be complex Hilbert spaces, and let $u : H \cong H'$ be a unitary operator (isometric linear isomorphism) with inverse $u^{-1}$. Let $\mu$ be a spectral measure on $H$. For any vectors $x, y \in H'$, the scalar complex measure of the conjugated spectral measure $u \mu u^{-1}$ associated with $x$ and $y$ is equal to the scalar complex measure of the original spectral measure $\mu$ associated with the transformed vectors $u^{-1}x$ and $u^{-1}y$. That is, for any measurable set $S$: $$\langle y, (u \mu(S) u^{-1}) x \rangle = \langle u^{-1}y, \mu(S) u^{-1}x \rangle$$

Let $H$ and $H'$ be complex Hilbert spaces, and let $u : H \cong H'$ be a unitary operator (an isometric linear isomorphism). Let $\mu$ be a spectral measure on a measurable space $\alpha$ taking values in the operators on $H$. For any vector $x \in H'$, the diagonal measure associated with the conjugated spectral measure $u \mu u^{-1}$ and the vector $x$ is equal to the diagonal measure associated with the original spectral measure $\mu$ and the vector $u^{-1}x$. That is, $$(u \mu u^{-1})_x = \mu_{u^{-1}x}$$ where for any spectral measure $\nu$ and vector $v$, the diagonal measure $\nu_v$ is the positive measure defined by $\nu_v(S) = \langle v, \nu(S)v \rangle$ for measurable sets $S \subseteq \alpha$.