Entropy Numbers of Certain Summation Operators

Given nonnegative real sequences a = (αk)k∈Z and b = (βk)k∈Z we study the generated summation operator Sa,b(x) := ( αk [∑ l<k βlxl ]) k∈Z , x = (xk)k∈Z , regarded as a mapping from `p(Z) to `q(Z). We give necessary and sufficient conditions for the boundedness of Sa,b and prove optimal estimates for its entropy numbers relative to the summation properties of a and b. Our results are applied to the investigation of the behaviour of P (∑ k∈Z α k|W (tk)| < ε ) and P ( sup k∈Z αk|W (tk)| < ε ) as ε → 0, where (tk)k∈Z is some nondecreasing sequence in [0,∞) and (W (t))t≥0 denotes the Wiener process. 2000 Mathematics Subject Classification: Primary: 47B06. Secondary: 47B37, 60G15.