Absolutely Summing Operators Between Hardy Spaces

Abstract

The Hardy space Hp , 1 ≤ p ≤ ∞, is a subspace of L p that contain of all functions with Fourier series P∞ n=0 cne int. If a = (an) is a given vector, then the diagonal operator da : Hp → Hq is defined by da( P∞ n=0 cne inf ty) = P∞ n=0 ancne int . The absolutely summing operatoru : X → Y is a linear operator between Banach spaces. We say that u is p summing operator for 1 ≤ p ≤ ∞ if there is a constant c ≥ 0 such that regardless of the natural number m and regardless of the choice of x1, ..., xm in X we have, Xm i=1 kuxik p !1 p ≤ c.sup    Xm i=1 |φ(xi)| p !1 p , φ ∈ X 0 , kφk ≤ 1    (1) In this thesis, we consider the diagonal operator da between Hardy spaces Hp and Hq where 1 ≤ p, q ≤ ∞ and a is the sequence (an). In this thesis we find necessary and sufficient conditions for this diagonal operator to be 2 summing. We were able to prove that da : Hp → Hq is 2 summing if and only if a ∈ l 2 . After that, we prove that this operator is 1 summing if and only if a ∈ l 1 .