PhyslibAlpha.AlgebraicFramework.HilbertSpace.Unbounded.BoundedSelfAdjointData

Let $H$ be a complex Hilbert space and $A: H \to H$ be a bounded self-adjoint operator. This definition constructs the real-valued Weak Operator Topology (WOT) spectral measure associated with $A$. It is obtained by taking the spectral measure of $A$ (viewed as a normal operator on the complex plane $\mathbb{C}$) and pushing it forward along the real-part function $\text{Re}: \mathbb{C} \to \mathbb{R}$. Because $A$ is self-adjoint, its spectrum $\sigma(A)$ is contained in $\mathbb{R}$, and this measure represents the resolution of the identity for $A$ over the real line.

Let $H$ be a complex Hilbert space and $A: H \to H$ be a bounded self-adjoint operator. Let $E_A$ denote the spectral measure of $A$ on the real line $\mathbb{R}$ (the resolution of the identity). For any vectors $x, y \in H$, the integral of the identity function $f(r) = r$ with respect to the complex-valued measure $\mu_{x,y}(\Delta) = \langle y, E_A(\Delta)x \rangle$ is equal to the inner product of $y$ and $Ax$: $$\int_{\mathbb{R}} r \, d\langle y, E_A(r) x \rangle = \langle y, Ax \rangle$$

Let $H$ be a complex Hilbert space and $A: H \to H$ be a bounded self-adjoint operator. Let $P_A$ denote the real-valued spectral measure (resolution of the identity) associated with $A$. If $T: H \to H$ is a unitary operator that commutes with $A$ (i.e., $AT = TA$), then for any Borel set $E \subseteq \mathbb{R}$, the spectral projection $P_A(E)$ commutes with $T$: $$P_A(E) T = T P_A(E)$$ Note that because $A$ is self-adjoint, the condition $AT=TA$ is sufficient to imply that $T$ commutes with the entire functional calculus of $A$ without requiring a separate hypothesis on the adjoint $A^*$.

Let $H$ be a complex Hilbert space and $A: H \to H$ be a bounded self-adjoint operator. There exists a real constant $C$ such that the spectral measure associated with $A$ on the real line $\mathbb{R}$ has bounded spectral support.