### 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
.

### Description:

CD , no of pages 48, 31021 , mathematics 2/2019