Monotonicity of hybrid operator norms for nested subgroups

Determine whether, for nested subgroups Γ ⊂ Λ ⊂ G, the hybrid operator norm associated to the pair (G,Γ) is bounded above by the hybrid operator norm associated to the pair (G,Λ) for all finitely supported functions f ∈ ℂG; that is, establish whether the inequality ∥f∥_{h,(G,Γ)} ≤ ∥f∥_{h,(G,Λ)} holds uniformly in f.

Background

The paper introduces the hybrid (2,1)-norm and the associated hybrid operator norm ∥·∥_h defined for a pair (G,H). It is shown that when Γ ⊂ Λ ⊂ G, the hybrid (2,1)-norm satisfies a natural monotonicity (bounded above) with respect to enlarging the subgroup. However, the authors point out that the corresponding statement for the hybrid operator norms is not established.

Resolving this would clarify how the operator norm completion B^*_r(G,H) behaves under inclusion of subgroups and would strengthen several stability results discussed later in the paper.