eu ON LINEAR AND MULTILINEAR MAPPINGS OF NUCLEAR TYPE

In this work, a technique is used in order to demonstrate the following lemmas “If s ∈ F(E,C(K)), then N0 ∞(S) = ‖S‖” and “If S ∈ F(E,Lp(Ω, μ)), then K 0 p(S) = ‖S‖” and open question is answered in 18.1.16 which belong to Albert Pietsch, at the same time Lemma 2.1 of B. M. Cerna is demonstrated consequently additional results has been got. Dedicated to the memory of our father: B. Cerna Figueroa AMS Subject Classification: 47H60, 46G25, 47L20 Following the ideas of Pietsch, I couldn’t extend the lemmas “If s ∈ F(E, C(K)), then N0 ∞(S) = ‖S‖” and “If S ∈ F(E,Lp(Ω, μ)), then K 0 p (S) = ‖S‖” toward the multilinear case. According to my interests fortunately, I could get Received: February 20, 2014 c © 2014 Academic Publications, Ltd. url: www.acadpubl.eu §Correspondence author 16 B.M.C. Maguiña, J.M. Ramos, H.F.C. Maguiña one technique based If f ∈ X∗ where X is a normed space and f 6= 0, then f is surjective and X = ker f ⊕M , where M is a one-dimensional subspace. These result and the theorem of Hahn Banach let me demonstrate the lemmas mentioned before, furthermore respond the question that is formulate in the book of Piestch in the lemma 18.1.16. furthermore this technique must have been extended to the multilinear case which let me get additional results. A problem arise that It’s more natural, There exist some relation between the theory that It served in the show of this results? We introduce the notations in the present work, for Banach spaces E1, · · · , En and F over the field K (R or C), we denote L(E1, · · · , En;F ) to the Banach space of all multi-linear and continuous applications of E1 × · · · × En over F with a natural norm given by ‖T‖ = sup xi∈BEi i=1,··· ,n ‖T (x1, · · · , xn)‖ where BEi denote the unitary ball of Ei, centered in 0. E ∗ k denote the dual topological of Ek, k = 1, · · · , n. For s ∈ 〈0,+∞〉 we denote by ls(F ) (or ls, F = K), the vector space of all sequences (yj) ∞ j=1 of elements that belong to F such that ls(yj) = ∥(yj)∞j=1 ∥∥ s =   ∞ ∑ j=1 ‖yj‖ s   1/s < +∞. For s ≥ 1, ‖.‖s is a norm, and for s < 1, is a s-norm. In any case we have a complete metric vector space. We denote by lw s (F ) the vector space of all sequences (yj) ∞ j=1 of elements that belong to F such that ∥∥∥(yj)∞j=1 ∥∥ w,s = ws(yj) = sup φ∈BF∗ ∥∥∥(φ(yj))∞j=1 ∥∥ s < +∞, so, (lw s (F ), ‖.‖w,s) is a metric vector space. For s = +∞, we consider l∞(F ) = l w ∞(F ) as a Banach space of all sequences (yj) ∞ j=1 of elements of F under the norm w∞(yj) = ∥∥∥(yj)∞j=1 ∥∥ ∞ = ∥∥∥(yj)∞j=1 ∥∥ w,∞ = sup j∈N ‖yj‖ ON LINEAR AND MULTILINEAR MAPPINGS... 17 Let 0 < r ≤ ∞, 1 ≤ p, q ≤ ∞, and 1 + 1r ≥ 1 p + 1 q . An operator s ∈ L(E,F ) is called (r, p, q)-nuclear if s = ∞ ∑ i=1 σiai ⊗ yi con (σj) ∞ j=1 ∈ lr, (ai) ∞ i=1 ∈ wq′(E ∗), and (yi) ∞ i=1 ∈ wp′(F ). In the case r = ∞ let us suppose that (σi) ∈ c0. We put N(r,p,q)(s) := inf lr(σi)wq′(ai)wp′(yi) (1) where the infimun is taken over all so-called (r, p, q)-nuclear representations described above. The class of all (r, p, q)-nuclear operators is denoted by N(r,p,q). F(E,F ) := Ideal of finite operators of finite range from E onto F . For every operators s ∈ F(E,F ) we put N (r,p,q)(s) = Nf,(r,p,q)(s) := inf lr(σi)wq′(ai)wp′(yi) (2) where the infimum is taken over all finite representations

one technique based If f ∈ X * where X is a normed space and f = 0, then f is surjective and X = ker f ⊕ M , where M is a one-dimensional subspace.These result and the theorem of Hahn Banach let me demonstrate the lemmas mentioned before, furthermore respond the question that is formulate in the book of Piestch in the lemma 18. 1.16.furthermore this technique must have been extended to the multilinear case which let me get additional results.
A problem arise that It's more natural, There exist some relation between the theory that It served in the show of this results?
We introduce the notations in the present work, for Banach spaces E 1 , • • • , E n and F over the field K (R or C), we denote L(E 1 , • • • , E n ; F ) to the Banach space of all multi-linear and continuous applications of E 1 × • • • × E n over F with a natural norm given by T = sup where For s ∈ 0, +∞ we denote by l s (F ) (or l s , F = K), the vector space of all sequences (y j ) ∞ j=1 of elements that belong to F such that For s ≥ 1, .s is a norm, and for s < 1, is a s-norm.In any case we have a complete metric vector space.We denote by l w s (F ) the vector space of all sequences (y j ) ∞ j=1 of elements that belong to F such that < +∞, so, (l w s (F ), .w,s ) is a metric vector space.
For s = +∞, we consider l ∞ (F ) = l w ∞ (F ) as a Banach space of all sequences (y j ) ∞ j=1 of elements of F under the norm where the infimun is taken over all so-called (r, p, q)-nuclear representations described above.
The class of all (r, p, q)-nuclear operators is denoted by N (r,p,q) .F(E, F ) := Ideal of finite operators of finite range from E onto F .For every operators s ∈ F(E, F ) we put N 0 (r,p,q) (s) = N f,(r,p,q) (s) := inf l r (σ i )w q ′ (a i )w p ′ (y i ) where the infimum is taken over all finite representations In the case r = +∞ the condition for (σ k ) ∞ k=1 is to be in c 0 .
The set of such applications satisfying such definition is a vector space and is denote by where the infimum is taken over all possible representations of T described in (3).
where the infimum is taken over all finite representations 1. Linear Operators of (r, p, q)-Nuclear Type In this section using the proposition 1.1 and one consequence of the Hanhn-Banach Theorem see [2] demonstrate the lemmas 19.2.5, 19.3.6 and to get similar result of 18.1.16.See [6].
of ( 4) and ( 5) we have where Proof.It is clear that for 1 p + 1 q = 1, one has from ( 1) and ( 2) Moreover it is clear that M < +∞.From the proposition (1.1) there exist Therefore, from the relation (9) we have that, one has ǫ > 0, Therefore x ≤ 1, from the relation (8) one has In addition, let Moreover, as consequence of the Hahn-Banach theorem (see [2]) there exists φ such that and further can choose σ k 0 such that Taking into account these last relations in the equation ( 11) we can get then for a given ǫ > 0 one has therefore, from the relations (12) one has: We know that From the last relation an the equation ( 13) one obtains , for all ǫ > 0, and ǫ > 0 ( Therefore, from the relations (10) y (14) one has From equations ( 7) and (15) one has required result.For p = +∞, we have where except for a set of measure zero } also we have: for ǫ > 0, there exists A > 0 and N ⊂ Ω with µ(N ) = 0 such that From the equations ( 4) and ( 16) one can get From the proposition (1.1) there exists x ∈ E such that Then it is clear that M < +∞ and of (17) we have Let In addition, let Moreover, as a consequence of the Hahn-Banach theorem (see [2]), there exists ϕ such that Therefore, from the relations (20) one has: we know that For ǫ > 0 we have that From the relations (21) and ( 22) we have: , for all ǫ > 0, and ǫ > 0 Therefore, from the last relations and (18) we have From equations ( 5) and (23) one has the required result.
In the Subsection 18.1.16Pietsch say for Special exponents the above result holds without any assumption on the underlying Banach space.Proposition 1.2.[6] N 0 (r,2,q) (s) = N (r,2,q) (s) for all s ∈ F(E, F ).
Proof.See lemma (1.1) on this article.

Multilinear Mappings of Nuclear Type
The next results is slightly different from the one given in lemma (1.1) and its proof can be performed following the lines of this reference.