The Heisenberg group associated to a Hilbert space

Let (X,〈⋅〉)(X,〈⋅〉) be a complex Hilbert space. The set HX=X×RHX=X×R equipped with the binary operation (x1,t1)⋅(x2,t2)=(x1+x2,t1+t2+2Im(〈x1,x2〉)) is the famous Heisenberg group. For all α>0α>0, k>0k>0 let Nα,k:HX→[0,∞)Nα,k:HX→[0,∞) be defined by Nα,k(x,t)=(‖x‖αk+|t|α2k)1α. We prove thatNα,k((x1,t1)⋅(x2,t2))≤([Nα,k(x1,t1)]k+[Nα,k(x2,t2)]k)1k if and only if α≥4kα≥4k. A similar result is proved for a real Hilbert space. Related questions are investigated.