Let $H \le G$ be an inclusion of countable, discrete groups, and let $\pi$ be unitary representation of $G$ on some Hilbert space $\mathcal{K}$. Is it possible to determine the orthogonal complete $(\mathcal{K}^{H})^{\perp}$, where $\mathcal{K}^{H} := \{ \xi \in \mathcal{K} : \pi(h) \xi = \xi ~~ \forall h \in H \}$ denotes the closed subspace of $\pi(H)$-invariant vectors in $\mathcal{K}$? That is, does this orthogonal complement admit another description aside from the one stating that $(\mathcal{K}^H)^\perp$ consists of all those vectors which are orthogonal to all the vectors in $\mathcal{K}^H$?
2026-04-13 02:52:53.1776048773
Orthogonal Complement of the $\pi(H)$-invariant Vectors in a Hilbert Space
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