The approximation numbers of Hardy – type operators on trees

The approximation numbers of Hardy–type operators on trees. Abstract The Hardy operator Ta on a tree Γ is defined by (Taf)(x) := v(x) x a u(t)f (t)dt for a, x ∈ Γ. Properties of Ta as a map from L p (Γ) into itself are established for 1 ≤ p ≤ ∞. The main result is that, with appropriate assumptions on u and v, the approximation numbers an(Ta) of Ta satisfy (*) lim n→∞ nan(Ta) = αp Γ |uv|dt for a specified constant αp and 1 < p < ∞. This extends results of Naimark, Newman and Solomyak for p = 2. Hitherto, for p = 2, (*) was unknown even when Γ is an interval. Also, upper and lower estimates for the l q and weak-l q norms of {an(Ta)} are determined.