PhyslibAlpha.QuantumMechanics.HilbertSpaces.FiniteTarget.Operators

Let $\mathcal{H}_d$ denote a $d$-dimensional complex Hilbert space. Let $\mathcal{B}(\mathcal{H}_d)$ be the space of bounded complex-linear operators on $\mathcal{H}_d$. For any real number $r \in \mathbb{R}$ and any operators $A, B \in \mathcal{B}(\mathcal{H}_d)$, the following associativity property between real scalar multiplication and operator composition holds: $$(r \cdot A) \circ B = r \cdot (A \circ B)$$ where $\circ$ denotes the composition of operators.

In the context of a finite-dimensional complex Hilbert space $\mathcal{H}_d$ (represented as $\mathbb{C}^d$), the action of real scalar multiplication commutes with the composition of bounded complex-linear operators. Specifically, for any real number $r \in \mathbb{R}$ and any two bounded complex-linear operators $A, B: \mathcal{H}_d \to \mathcal{H}_d$, the relation $r \cdot (A \circ B) = A \circ (r \cdot B)$ holds.

Let $\mathcal{H}_d$ be the $d$-dimensional complex Hilbert space $\mathbb{C}^d$. The space of bounded linear operators on this space, denoted by $\mathcal{B}(\mathcal{H}_d)$ (or $\mathcal{H}_d \to_{\text{L}}[\mathbb{C}] \mathcal{H}_d$), admits a decomposition for self-adjoint operators. This means that every self-adjoint operator $A \in \mathcal{B}(\mathcal{H}_d)$ can be uniquely expressed as the difference of two positive operators, $A = A^+ - A^-$, where $A^+$ and $A^-$ are the positive and negative parts of $A$ obtained via the continuous functional calculus.