ON SOME INEQUALITIES FOR THE APPROXIMATION NUMBERS OF THE SUM AND PRODUCT OF OPERATORS

. We prove the inequalities:

where {an(S)} is the sequence of the approximation numbers of the linear and bounded operators S : X → X (S ∈ L(X)).X is a Banach space.

INTRODUCTION
Let H be a Hilbert space and let S : H → H, S ∈ L(H), be a compact linear operator.We denote by s n (S) = λ n (SS * ) 1 2 , where λ n (SS * ) is the sequence of the eigenvalues of (SS * ) 1 2 , ordered decreasing and repeating each one as many times as its algebraic multiplicity.
In this way we can consider X a Banach space and S ∈ L(X).Then, the approximation numbers of S, (a n (S)), are defined as follows: By using the inequalities (a) and (b), for m = n, in [6], are obtained some inequalities of the above type ( 1) and ( 2), but in the right side appear the factor 2, because By reiteration we obtain: An analog inequality is true for the sum of r operators (S i ∈ L(X)).The purpose of this paper is to prove, in a simple way, that the factor 2 r−1 can be replaced by r.Some applications are also presented.
The inequality (3) results in a same way by using the inequality (a).
By means of these inequalities we obtain: Theorem 2. For the operators S i ∈ L(X), i = 1, 2, ...r, the following inequalities hold: Proof.We prove only ( 6) and ( 7) since the proof of ( 5) is similar.For to prove (6) we use the inequality (4) and the fact that the sequence (a n (S)) is decreasing.Then we obtain: The proof is fulfiled.
Remark 1.These inequalities are also true for all additive and multiplicative s-number sequences and also for the dyadic entropy numbers.

S i φ ≤ r 1 S
In the case of Hilbert spaces, we obtain r i i φ , by using the inequality (1) and reiteration.