P-Quasi-Λ-Nuclear Operators Between Locally Convex Spaces * ﺩ

Al-Aqsa University Journal (Natural Sciences Series) , Vol.14, No.2, Pages 18-26, Jan. 2010 ISSN 2070-3155

p-Quasi-λ-Nuclear Operators Between Locally Convex Spaces * ﺩ. ﺯﻴﺎﺩ ﺼﺎﻓﻲ * ﺩ. ﺃﺤﻤﺩ ﺍﻻﺸﻘﺭ

ﺍﻟﻤﻠﺨﺹ ﻓﻲ ﻫﺫﺍ ﺍﻟﺒﺤﺙ ﻗﻤﻨﺎ ﺒﺘﻌﻤﻴﻡ ﻤﻔﻬﻭﻡ ﺍﻟﻤﺅﺜﺭﺍﺕ 2- ﺸﺒﻪ-λ- ﺍﻟﻨﻭﻭﻴﺔ ﺒﻴﻥ ﺍﻟﻔﻀﺎﺀﺍﺕ ﺍﻟﻤﻌﻴﺎﺭﻴﺔ (λ⊆ l1) ﺇﻟﻰ p- ﺸﺒﻪ-λ- ﺍﻟﻨﻭﻭﻴﺔ ﺒﻴﻥ ﺍﻟﻔﻀﺎﺀﺍﺕ ﺍﻟﻤﺤﺩﺒﺔ ﺍﻟﻤﺤﻠﻴﺔ ( ∞p >0, λ⊆ l)، ﻭﻜﺫﺍﻟﻙ ﺩﺭﺴﻨﺎ ﺍﻟﻌﻼﻗﺔ ﺒﻴﻥ ﺍﻟﻤﺅﺜﺭﺍﺕ p- ﺸﺒﻪ-λ- ﺍﻟﻨﻭﻭﻴﺔ، ﻭﺍﻟﻤﺅﺜﺭﺍﺕ ﺍﻟﻨﻭﻭﻴﺔ، ﻭﺍﻟﻤﺅﺜﺭﺍﺕ λ- ﺍﻟﻨﻭﻭﻴﺔ، ﻭﺍﻟﻤﺅﺜﺭﺍﺕ ﺸﺒﻪ-λ-ﺍﻟﻨﻭﻭﻴﺔ، ﻭﺒﺭﻫﻨﺎ ﺃﻥ ﺘﺭﻜﻴﺏ ﻤﺅﺜﺭﻴﻥ ﺃﺤﺩﻫﻤﺎ p- ﺸﺒﻪ-λ- ﻨﻭﻭﻱ ﻴﻜﻭﻥ p - ﺸﺒﻪ-λ- ﻨﻭﻭﻱ .

ABSTRACT

In this paper we generalize the concept of 2-quasi-λ-nuclear operators between Normed spaces (λ⊆ l1) to p-quasi-λ-nuclear operators between locally convex spaces (p >0, λ⊆ l∞ ) and we study the relationship between p-quasi-λ-nuclear, nuclear operators, λ-nuclear, quasi-nuclear and quasi-λ- nuclear. Also we prove that the composition of two operators, one of them is a p-quasi-λ-nuclear, is again a p-quasi-λ-nuclear operator.

* ﻗﺴﻡ ﺍﻟﺭﻴﺎﻀﻴﺎﺕ – ﻜﻠﻴﺔ ﺍﻟﻌﻠﻭﻡ- ﺠﺎﻤﻌﺔ ﺍﻷﻗﺼﻰ - ﻏﺯﺓ - ﻓﻠﺴﻁﻴﻥ.

p-Quasi-λ-Nuclear Operators Between Locally…

1. Preliminary. By Shatanawi [5], the operator T from a normed space E into a normed space F is said to be 2-quasi-λ-nuclear if there is a sequence

()α n ∈ λ ( λ ⊆ l1 ) and a bounded sequence ()an in E ′ such that ∞ 12 ⎛⎞2 Tx≤〈〉∞∀∈⎜⎟∑ |α nn || x , a | < , x E . ⎝⎠n =1 In this paper, we generalize this definition to p-quasi-λ-nuclear operator between locally convex spaces where λ ⊆ l ∞ and p > 0 . By locally convex spaces E, we mean a locally convex spaces E with a topology induced by a continuous sequence of seminorms. Throughout this paper E, F,… will denote locally convex spaces over the same field K. By E' we mean the set of all continuous linear operators from E into K. We let 〈〉=x ,()aax for all x in E and a in E'.

By lp [respectively, c0, l∞], we mean the usual Banach space of all scalar- valued, p-power summable [respectively, zero-convergent, bounded] sequences. By λ we mean any sequence space subset of l ∞ .

We say that the sequence aa= ()n dominates the sequence bb= ()n , written bOann= (), if there exists a real number M > 0 such that

bMann≤ for all n ∈ . A set A of sequences of non-negative real numbers is called a Köthe set, if it satisfies the following conditions:

(1) ∀∈ab, A, there is cA∈ with aOcnn= () and bOcnn= ().

(2) ∀∈r there exists aA∈ with ar > 0 . The sequence space λ()A defined by ⎧⎫∞ λ()Axxqxxa==⎨⎬ (na ):() =∑ | nn | <∞ ⎩⎭n =1 is called a Köthe space generated by A (see [2]). The space s of rapidly decreasing sequences is a Köthe space which is k generated by the set Ank=∈{( ): }, so

Zeyad safi, Ahmed al-Ashqar, J. Al-Aqsa Unv., Vol.14, No.2, 2010

∞ ⎧⎫k snk=<∞∀∈⎨⎬():ααnn∑ (see [3]). ⎩⎭n =1 For each closed and absolutely convex bounded subset B of a locally convex space E

E B =∈{xEx : ∈ρ B for some ρρ >= 0} U B ρ>0 is a linear subspace of E which is the linear span of B and B is absorbent in E B . Also we have rEBB: → defined by

rxB ()=>∈ inf{ρ 0: xρ B } is a norm, so we shall always consider E B as a normed space with respect to the norm rB . We call the disk B a

Banach disk if E B happens to be a Banach space (see [1]). 2. Main Results. Definition 2.1 [1] A continuous operator TE: → F is said to be nuclear iff there is a sequence ()α n ∈ l1 , an equicontinuous sequence

()an in E ′ , a Banach disk B in F and a bounded sequence ()y n in FB such that ∞ Tx=〈〉∀∈∑α nnn x, a y , x E . n =1 Definition 2.2 A continuous operator TE: → Fis said to be p-quasi- nuclear iff for each continuous seminorm q on F there is an equicontinuous sequence ()an in E ′ such that ∞ 1 p ⎛⎞p qTx( )≤〈〉⎜⎟∑ | x , an | < ∞∀∈ , x E . ⎝⎠n =1 Definition 2.3 A continuous operator TE: → F is said to be p-quasi-λ- nuclear iff for each continuous seminorm q on F there is a sequence

()α n ∈ λ and an equicontinuous sequence ()an in E ′ such that ∞ 1 p ⎛⎞p qTx()≤〈〉∞∀∈⎜⎟∑ |α nn ||, x a | < , x E . ⎝⎠n =1

p-Quasi-λ-Nuclear Operators Between Locally…

The operator TE: → F is said to be quasi-nuclear (respectively, quasi- λ-nuclear) if it is 1-quasi-nuclear ( respectively, 1-quasi-λ-nuclear) Note: If (,||||)F ⋅ is a normed space, then TE: → F is called a p- quasi-λ-nuclear operator if there is a sequence ()α n in λ and a bounded sequence ()an in E ′ such that ∞ 1 p ⎛⎞p ||Tx ||≤〈〉∀∈⎜⎟∑ |α nn || x , a | , x E . ⎝⎠n =1

Proposition 2.4 If λ = l1 and p > 1 , then each quasi-λ-nuclear operator is p-quasi-λ-nuclear.

Proof: Let TE: → F be a quasi-λ-nuclear operator. Then for each continuous seminorm q on F, there exist an ()α n ∈ λ and an equicontinuous sequence ()an in E ′ such that ∞ qTx()≤〈〉∀∈∑ |α nn ||, x a | , x E . n =1 Let . Since p > 1 , there exists r > 1 such that 11+=1, β = ∑||α n pr n and so by Hölder's inequality we have ∞∞1 1 11+ ⎛⎞ qTx( )≤〈〉 |ααα |pr | x , a |= (| |)r (| |)p | 〈〉 x , a | ∑∑nn⎜⎟ nnn nn==11⎝⎠ 11 ∞∞rp ⎛⎞⎛p ⎞ ≤〈〉⎜⎟⎜∑∑ |ααnnn | | ||xa , | ⎟ ⎝⎠⎝nn==11 ⎠ 1 1 p ∞∞⎛⎞1 p 1 rp⎛⎞ r p =βα⎜⎟∑∑ |nn ||〈〉xa , | =⎜⎟ | αβ n || 〈 x , a n 〉 | . ⎝⎠nn==11⎝⎠

Zeyad safi, Ahmed al-Ashqar, J. Al-Aqsa Unv., Vol.14, No.2, 2010

1 r Since ()α n ∈ λ and ()β an is an equicontinuous sequence in E ′ , T is a p-quasi-λ-nuclear operator. Proposition 2.5 If p > 1 , then every p-quasi-nuclear operator is a p- quasi- l p -nuclear operator. Proof: Assume TE: → F is p-quasi-nuclear operator. Then for each continuous seminorm q on F there exists an equicontinuous sequence

()an in E ′ such that ∀∈x E we have 1 p 1 p p ⎛⎞∞∞⎛⎞a p ⎜⎟n qTx()≤〈〉 |, x ann | = |, 〈〉 x a | x , , ⎜⎟∑∑⎜⎟|,〈〉xa |1 p ⎝⎠nn==11⎝⎠n without lose of generality we can assume that 〈xa,0n 〉≠ . 1 p Since (,〈〉∈x alnp ) and (|,|)axann〈〉 is an equicontinuous sequence in E ′ , T is a p-quasi- l p -nuclear operator. Proposition 2.6 If p >1 , then every p-quasi-λ-nuclear operator is a p- quasi-nuclear operator. Proof: Let TE: → Fbe a quasi-λ-nuclear operator. Then for every continuous seminorm q on F there exist a sequence ()αn ∈λ , and an equicontinuous sequence ()an in E ′ such that ∞ 1 p ⎛⎞p qT()() x≤〈〉∀∈⎜⎟∑ |αnn ||, xa | , x E . ⎝⎠n

1 p ∞ p ⎛⎞1 p = ⎜⎟∑ xa ,|αnn | . ⎝⎠n 1 p Since (|αnn |a ) is an equicontinuous sequence in E ′ , T is p-quasi- nuclear operator. Theorem 2.7 If λ = l1 and p >1 , then, then every nuclear operator is p-quasi-λ-nuclear operator.

p-Quasi-λ-Nuclear Operators Between Locally…

Proof: Let TE: → Fbe a quasi-λ-nuclear operator. Then there exist a sequence ()α n ∈ λ , an equicontinuous sequence ()an in E ′ , a Banach disk B in F, and a bounded sequence ()y n in FB such that for all x ∈ E ∞ Tx()= ∑α nnn〈〉 xa , y . n =1 Then for all continuous seminorms q on F and x ∈ E we have ⎛⎞∞ qT(()) x=〈〉 q⎜⎟∑α nnn x , a y ⎝⎠n =1 ∞ ≤〈〉∑ |α nnn ||x ,aqy | ( ). n =1

Since ()y n is a bounded sequence, there exists M > 0 such that

qy()n ≤ M, and so ∞ qT(()) x≤〈〉∑ |α nn ||, x a | M n =1 ∞ = ∑ |α nn ||〈〉xMa , |. n =1 Thus, T is a quasi-λ-nuclear operator, and by proposition (2.4) T is a p- quasi-λ-nuclear operator. Theorem 2.8 Let 0 <<<∞pq . Then

(1) Every p-quasi- l1 -nuclear operator is q-quasi- l1 -nuclear operator. (2) The continuous operator TE: → F is p-quasi-s-nuclear iff it is q-quasi-s-nuclear. Proof:(1) Suppose that 0 <<<∞pq and TE: → F is a p-quasi-λ- nuclear operator. Then for each continuous seminorm r on F there exists an

()α n ∈ l1 and an equicontinuous sequence ()an in E ′ such that

Zeyad safi, Ahmed al-Ashqar, J. Al-Aqsa Unv., Vol.14, No.2, 2010

1 ⎛⎞∞ p p rTx()≤〈〉∀∈⎜⎟∑ |α nn ||, x a | x E . ⎝⎠n =1 q 1 qp− p 11 Let t = qp− , then tq==−1 q, so tqp+ =1, then we have ∞ p 1 1 rTx()≤〈〉∀∈ |||||,|ααt qp x ap x E . ()∑ nn( n) n =1 By Hölder's inequality we have 1 1 q qp ∞∞t q 1 t p p p ⎛⎞⎛⎞p rTx()≤〈〉∀∈ ||ααt ||q |,| x ap , x E . ()⎜⎟∑∑()nnn⎜⎟()() ⎝⎠nn==11⎝⎠ Now we have ∞∞11pt q ⎛⎞⎛q ⎞ rTx()≤ ⎜⎟⎜∑∑ ||ααnnn |||,|〈〉 x a ⎟ ∀∈ x E . ⎝⎠⎝nn==11 ⎠ ∞ 1 pt Let ⎛⎞and ba= β . Then βα= ⎜⎟∑||n nn ⎝⎠n =1 ∞ 1 q ⎛⎞q rTx()≤〈〉∀∈⎜⎟∑ |α nn ||, x b | x E , ⎝⎠n =1 and so T is q-quasi- l1 -nuclear operator. (2) The '' if " part condition follows from part (1). To prove the only ''if part", let TE: → F be a q-quasi-s-nuclear operator. Then for each continuous seminorm r on F, there exist an ()α n ∈s and an equicontinuous sequence ()an in E ′ such that 1 ∞ q ⎛⎞q rTx()≤〈〉∀∈⎜⎟∑ |α nn ||, x a | x E . ⎝⎠n =1 Then by Jeansen’s inequality [4], we have 1 ⎛⎞∞ p p rTx()≤〈〉∀∈ |α ||,q x a |p x E . ⎜⎟∑ nn ⎝⎠n =1 24

p-Quasi-λ-Nuclear Operators Between Locally…

pq To finish our proof, it is enough to show that (|α n | )∈s . Since k ()α n ∈s , then ()α n n is bounded for all k ∈ . Let h be a natural p number such that hk.2≥+. By hypothesis, there is a positive number q h M with α n nM≤∀∈ n . Then ppp p h h qqqq ()ααnnnnMn=≤∀∈ . Now for a fixed k ∈ we have pp pph k +2 qqqq ααnnnnMn≤≤∀∈ ,

p p k −2 q q hence α n nMnn≤∀∈ , and so ∞∞p p k −2 q q ∑∑α n nM≤ n<∞. nn=1 pq Therefore, (|α n | )∈s , hence T is a p-quasi-s-nuclear operator. Theorem 2.9 Let TE: → Fand SF: → Gbe any continuous operators. If either T or S is a p-quasi-λ-nuclear operator, then STo is a p-quasi-λ-nuclear operator. Proof: Suppose SF: → G is a p-quasi-λ-nuclear operator. Then for each continuous seminorm q on F there exist ()α n ∈ λ and an equicontinuous sequence ()bn in F ′ such that ∞ 1 p ⎛⎞p qSy()≤〈〉∀∈⎜⎟∑ |α nn ||, yb | , y F . ⎝⎠n =1 Since Tx∈ F for all x ∈ E , we have

Zeyad safi, Ahmed al-Ashqar, J. Al-Aqsa Unv., Vol.14, No.2, 2010

∞ 1 p ⎛⎞p qSTx()≤〈〉⎜⎟∑ |||,|α nn Txb ⎝⎠n =1 1 p ⎛⎞∞ ′ p =⎜⎟∑ |α nn ||〈〉∀∈x ,Tb ( ) | , x E , ⎝⎠n =1 ′ By [2], (())Tbn is an equicontinuous sequence in E', and so STo is a p- quasi-λ-nuclear operator. Now let TE: → Fbe any p-quasi-λ-nuclear operator and let q be any continuous seminorm on G Since qSo is a seminorm on F. Then there exist a sequence ()α n in λ and an equicontinuous sequence ()an in E ′ such that ∞ 1 p ⎛⎞p ()()((()))|||,|qSTxo =≤〈〉∀∈ qSTx⎜⎟∑ α nn xa x E , ⎝⎠n =1 and so STo is a p-quasi-λ-nuclear operator.

References [1] Jarchow, H. “Locally Convex Spaces”, B. G. Teubner Stuttgart, (1981). [2] Pietsch, A. “Nuclear Locally Convex Spaces”, Akademic Verlin, Berlin, (1972). [3] Randtke Dan: Characterization of Precompact Maps, Schwartz Spaces and Nuclear Spaces, Amer. Math. Soc. 165, 87-101 (1972). [4] Rudin, W. “Real and Complex Analysis”, McGraw-Hill (1987). [5] Shatanawi,W.: 2-Quasi-λ-Nuclear Maps, Turk.J.Math., 29, 157-167 (2005).