IntertwiningOperator - crowlogic/arb4j GitHub Wiki

An intertwining unitary operator is a specific type of linear operator that arises in the context of Hilbert spaces and representation theory. To define it precisely, let's introduce some necessary concepts:

• Hilbert Space: A complete inner product space, which is a vector space equipped with an inner product that is complete with respect to the norm induced by the inner product.
• Unitary Operator: An operator $U$ between two Hilbert spaces $H_1$ and $H_2$ is called unitary if it preserves the inner product. That is, for all $x, y$ in $H_1$, it holds that $\langle Ux, Uy \rangle_{H_2} = \langle x, y \rangle_{H_1}$, and $U$ is surjective.
• Representation: A representation of a group $G$ on a Hilbert space $H$ is a homomorphism from $G$ to the group of bounded operators on $H$, typically denoted as $\pi: G \rightarrow B(H)$. This means that each element $g$ of the group $G$ is associated with an operator $\pi(g)$ in a way that respects the group operation.

An intertwining operator between two representations $(\pi_1, H_1)$ and $(\pi_2, H_2)$ of a group $G$ is a linear operator $T: H_1 \rightarrow H_2$ such that the action of $T$ respects the group action represented by $\pi_1$ and $\pi_2$. Specifically, $T$ intertwines $\pi_1$ and $\pi_2$ if for every $g$ in $G$, the following condition holds:

$$ T\pi_1(g) = \pi_2(g)T. $$

This condition means that applying the representation $\pi_1(g)$ to a vector in $H_1$ and then applying the operator $T$ yields the same result as first applying $T$ and then applying the representation $\pi_2(g)$ to the vector.

An intertwining unitary operator is an intertwining operator that is also unitary. That is, it not only satisfies the intertwining property with respect to two representations of a group but also preserves the inner product (making it an isomorphism of Hilbert spaces). Such operators are particularly important in the study of equivalent representations, harmonic analysis, and quantum mechanics, where they provide a means of relating different representations of symmetry groups in a manner that preserves the geometric structure of the Hilbert spaces involved.