2-SIDED ESTIMATES OF THE APPROXIMATION NUMBERS OF CERTAIN VOLTERRA-INTEGRAL-OPERATORS

We consider the Volterra integral operator T:L-P(R+)-->L-P(R+) defined (Tf)(x)=v(x)integral(0)(x) u(t)f(t) dt. Under suitable conditions on u. and v, upper and lower estimates for the approximation numbers a(n) (T) of T are established when 1<p<infinity. When p=2, these yield limn -->infinity na(n)(T)=pi(-1)integral(0)(infinity) \u(t)v(t)\dt. We also provide upper and lower estimates for the l(alpha) and weak l(alpha) norms of (a(n)(T)) when 1<alpha<infinity.