The Approximation Numbers of Hardy‐Type Operators on Trees

The Hardy operator T a on a tree Γ is defined by ( T a f ) ( x ) : = v ( x ) ∫ a x f ( t ) u ( t ) d t for  a , x ∈ Γ . Properties of T a 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 a n ( T a ) of T a satisfy( ∗ )lim n → ∞ n a n ( T a ) = α p ∫ Γ| u v |d tfor 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 l q and weak‐ l q norms of a n ( T a ) are determined. 2000 Mathematical Subject Classification : 47G10, 47B10.