The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T a on a tree Γ is defined by \[ (T_af)(x):=v(x) \int^x_a f(t)u(t)\,dt \quad \mbox{for } a, x\in \Gamma. \] Properties of T a as a map from Lp (Γ) into itself are established for 1 ≤ p ≤ ∞. The main result is that, with appropriate assumptions on u and v , the approximation numbers a n ( T a ) of T a satisfy \begin{equation*} \tag{$*$} \lim_{n\rightarrow \infty} na_n(T_a) = \alpha_p\int_{\Gamma}|uv|\,dt \end{equation*} for a specified constant α p 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 lq and weak- lq norms of a n ( T a ) are determined. 2000 Mathematical Subject Classification : 47G10, 47B10.