Relative property (T) for a pair and relative Kazhdan pairs

Let $G$ be a topological group and let $H\le G$ be any subgroup (Subgroup), not assumed normal or closed. The pair $(G,H)$ has relative property (T) if every strongly continuous unitary representation of $G$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners) with almost invariant vectors (Almost invariant vectors for a unitary representation) has a nonzero vector fixed by every $\pi (h)$ for $h\in H$.

A pair $(Q,\epsilon )$ with $Q\subseteq G$ and $\epsilon >0$ is a relative Kazhdan pair for $(G,H)$ if every strongly continuous unitary representation of $G$ having a $(Q,\epsilon )$-invariant unit vector has a nonzero $H$-invariant vector. A subset $Q\subseteq G$ is a relative Kazhdan set if $(Q,\epsilon )$ is a relative Kazhdan pair for some $\epsilon >0$.

When $H=G$, these are respectively Kazhdan's property (T), Kazhdan pairs, and Kazhdan sets (Kazhdan's property (T), Kazhdan pairs, Kazhdan sets and Kazhdan constants).

• The zero-space representation has no unit vectors and no almost invariant vectors, so it creates no exception to either implication.
• If $H=\{e\}$, every vector is $H$-invariant; hence $(G,\{e\})$ has relative property (T), and every $(Q,\epsilon )$ is a relative Kazhdan pair.
• The subgroup need not be normal for $H$-invariant vectors or for the relative pair definition to make sense. Closedness is a hypothesis in the cited KHV formulation, but the definitions stated here also make sense for a nonclosed subgroup.
• No local compactness, Hausdorffness, countability, or choice assumption is part of these definitions.