On operator ideals using weighted CesØ£ ro sequence

Journal of the Egyptian Mathematical Society (2013) xxx, xxx–xxx

Egyptian Mathematical Society

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On operator ideals using weighted Cesa`ro sequence space Amit Maji *, P.D. Srivastava Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, West Bengal, India Received 12 February 2013; revised 6 September 2013; accepted 8 October 2013

KEYWORDS s-Numbers; Approximation numbers; Operator ideals; Sequence space

Abstract Let s ¼ ðsn Þ be a sequence of s-numbers in the sense of Pietsch. In this paper we have introduced a class AðsÞ aro sequence space for p;q of s-type cesðp; qÞ operators by using weighted Ces forms a quasi-Banach operator ideal. Moreover, the 1 < p < 1. It is shown that the class AðsÞ p;q inclusion relations among the operator ideals as well as the inclusion relations among their duals are established. Finally, we have proved that the class AðaÞ p;q of approximation type cesðp; qÞ operators is small. 2010 MATHEMATICS SUBJECT CLASSIFICATION:

47B06; 47L20

ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00

1. Introduction Due to the immense applications in spectral theory, geometry of Banach spaces, theory of eigenvalue distributions etc., the theory of operator ideals occupies a special importance in * Corresponding author. Tel.: +91 9093929590. E-mail addresses: [email protected] (A. Maji), [email protected] (P.D. Srivastava). Peer review under responsibility of Egyptian Mathematical Society.

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functional analysis. Many useful operator ideals have been defined by using sequence of s-numbers. In 1963, Pietsch [1] introduced the approximation numbers of a bounded linear operator in Banach spaces. Subsequently, different s-numbers, namely Kolmogorov numbers, Gel’fand numbers, etc. are introduced to the Banach space setting. For the unifications of different s-numbers, Pietsch [2] defined an axiomatic theory of s-numbers in Banach spaces. aro sequence space cesp [3,4] is deFor 1 < p < 1, the Ces fined as ( !p ) 1 n X 1X cesp ¼ x ¼ ðxk Þ 2 w : jxk j < 1 ; n k¼1 n¼1

1110-256X ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2013.10.005

Please cite this article in press as: A. Maji, P.D. Srivastava, On operator ideals using weighted Cesa`ro sequence space, Journal of the Egyptian Mathematical Society (2013), http://dx.doi.org/10.1016/j.joems.2013.10.005

2

A. Maji, P.D. Srivastava

where w is the set of all real or complex sequences. The space cesp 1 1 p p P 1 Pn is complete with respect to the norm kxk ¼ jx j . k k¼1 n n¼1

It is easy to verify that if 1 < p 6 r < 1, then cesp # cesr . For more on the Cesaro sequence space and Cesaro operator one can refer [5–8]. Pietsch [1] defined an operator T 2 LðE; FÞ to be lp type oper1 P ðan ðTÞÞp is finite for 0 < p < 1, where ðan ðTÞÞ is the ator if n¼1

sequence of approximation numbers of the bounded linear operator T. Later on Constantin [9] generalized the class of lp type operators to the class of ces p type operators by using the Cesaro sequence spaces, where an operator T 2 LðE; FÞ is called 1 P p P n 1 ces p type if is finite, 1 < p < 1. For A p k¼1 ak ðTÞ n n¼1

type operators and Stolz mappings one can see [10,11]. The purpose of this paper is to study s-type cesðp; qÞ operators using weighted Cesaro sequence space. The s-type cesðp; qÞ operators are more general than the ces p type operators. We show that the class AðsÞ p;q of s-type cesðp; qÞ operators is a quasiBanach operator ideal. Moreover, the inclusion relations among the operator ideals as well as the inclusion relations among their duals are established. Finally, we also prove that the class AðaÞ p;q of approximation type cesðp; qÞ operators is small. 2. Preliminaries Throughout this paper we denote E; F as the real or complex Banach spaces and LðE; FÞ as the space of all bounded linear operators from E to F. Let L be the class of all bounded linear operators between arbitrary Banach spaces. We denote E0 as the dual of E and x0 is the continuous linear functional on E. N and Rþ stand for the set of all natural numbers and the set of all nonnegative real numbers respectively. Let x0 2 E0 and y 2 F. Then the map x0 y : E ! F is defined by ðx0 yÞðxÞ ¼ x0 ðxÞy; x 2 E. We now state few results which will be used in the sequel. Before it, we recall some basic definitions and terminologies of s-numbers of operators and operator ideals. Definition 2.1. A finite rank operator is a bounded linear operator whose dimension of the range space is finite. Definition 2.2 ([12,13]). A map s ¼ ðsn Þ : L ! Rþ assigning to every operator T 2 L a nonnegative scalar sequence ðsn ðTÞÞn2N is called an s-number sequence if the following conditions are satisfied: (S1) monotonicity: kT k ¼ s1 ðT Þ P s2 ðT Þ P P 0, for T 2 LðE; F Þ (S2) additivity: smþn1 ðS þ T Þ 6 sm ðSÞ þ sn ðT Þ, for S; T 2 LðE; F Þ; m; n 2 N (S3) ideal property: sn ðRST Þ 6 kRksn ðSÞkT k, for some R 2 LðF ; F 0 Þ; S 2 LðE; F Þ and T 2 LðE0 ; EÞ, where E0 ; F 0 are arbitrary Banach spaces (S4) rank property: If rankðT Þ 6 n then sn ðT Þ ¼ 0 (S5) norming property: sn ðI : ln2 ! ln2 Þ ¼ 1, where I denotes the identity operator on the n-dimensional Hilbert space ln2 .

We call sn ðTÞ the nth s-number of the operator T. For results on s-number sequence, refer [2,14–16]. We now give some examples of s-number sequences of a bounded linear operator. Let T 2 LðE; FÞ and n 2 N. The nth approximation number, denoted by an ðTÞ, is defined as an ðTÞ ¼ inf fkT Lk : L 2 LðE; FÞ; rankðLÞ < ng. The nth Gel’fand number, denoted by cn ðTÞ, is defined as cn ðTÞ ¼ inf fkTJM k : M E; codimðMÞ < ng, where JM : M ! E be the natural embedding from subspace M of E into E. The nth Kolmogorov number, denoted by dn ðTÞ, is defined as dn ðTÞ ¼ inf fkQN ðTÞk : N F; dimðNÞ < ng, where QN : E ! E=N be the quotient map from E onto E=N. The nth Weyl number, denoted by xn ðTÞ, is defined as xn ðTÞ ¼ inf fan ðTAÞ : jjAjj 6 1; where A : ‘2 ! Eg, where an ðTAÞ is an nth approximation number of the operator TA. The nth Chang number, denoted by yn ðTÞ, is defined as yn ðTÞ ¼ inf fan ðBTÞ : jjBjj 6 1; where B : F ! ‘2 g, where an ðBTÞ is an nth approximation number of the operator BT. The nth Hilbert number, denoted by hn ðTÞ, is defined as hn ðTÞ ¼ sup fan ðBTAÞ : jjBjj 6 1; jjAjj 6 1; where B : F ! ‘2 and A : ‘2 ! Eg. Remark 2.1 [16]. If T is compact and is defined on a Hilbert space, then all the s-numbers coincide with the singular values 1 of T i.e. the eigenvalues of jTj, where jTj ¼ ðT TÞ2 . Proposition 2.1 [16, p. 115]. Let T 2 LðE; FÞ. Then hn ðTÞ 6 xn ðTÞ 6 cn ðTÞ 6 an ðTÞ dn ðTÞ 6 an ðTÞ.

and

hn ðTÞ 6 yn ðTÞ 6

Definition 2.3 [16, p. 90]. An s-number sequence s ¼ ðsn Þ is called injective if, given any metric injection J 2 LðF; F0 Þ, sn ðTÞ ¼ sn ðJTÞ for all T 2 LðE; FÞ. Definition 2.4 [16, p. 95]. An s-number sequence s ¼ ðsn Þ is called surjective if, given any metric surjection Q 2 LðE0 ; EÞ; sn ðTÞ ¼ sn ðTQÞ for all T 2 LðE; FÞ. Proposition 2.2 [16, pp. 90–94]. The Gel’fand numbers and the Weyl numbers are injective. Proposition 2.3 [16, p. 95]. The Kolmogorov numbers and the Chang numbers are surjective. The following lemma is required to prove our theorems. Lemma 2.1 [2]. Let S; T 2 LðE; FÞ. Then jsn ðTÞ sn ðSÞj 6 kT Sk for n ¼ 1; 2; . . .. Definition 2.5 (Dual s-numbers [2]). For each s-number sequence s ¼ ðsn Þ, a dual s-number sequence sD ¼ ðsD n Þ is defined by 0 sD n ðTÞ ¼ sn ðT Þ

for all T 2 L, where T0 is the dual of T.


3 Definition 2.6 [14, p. 152]. An s-number sequence is called symmetric if sn ðTÞ P sn ðT0 Þ for all T 2 L. If sn ðTÞ ¼ sn ðT0 Þ then the s-number sequence is said to be completely symmetric.

Definition 2.10 [14, p. 68]. An operator ideal M is called symmetric if M M0 and is called completely symmetric if M ¼ M0 .

Theorem 2.1 [14, p. 152]. The approximation numbers are symmetric i.e. an ðT0 Þ 6 an ðTÞ for T 2 L.

3. s-type cesðp; qÞ operators

Theorem 2.2 [16, p. 95]. Let T 2 L. Then cn ðTÞ ¼ dn ðT0 Þ

and

cn ðT0 Þ 6 dn ðTÞ.

In addition, if T is a compact operator, then cn ðT0 Þ ¼ dn ðTÞ. Theorem 2.3 [16, p. 96]. Let T 2 L. Then 0

xn ðTÞ ¼ yn ðT Þ

and

0

yn ðTÞ ¼ xn ðT Þ.

Theorem 2.4 [16, p. 97]. The Hilbert numbers are completely symmetric i.e. hn ðTÞ ¼ hn ðT0 Þ for all T 2 L.

Let q ¼ ðqk Þ be Pa bounded sequence of positive real numbers. aro seDefine Qn ¼ nk¼1 qk ; n 2 N. Then the weighted Ces quence space cesðp; qÞ; 1 < p < 1 is defined by ( !p ) 1 n X 1 X cesðp; qÞ ¼ x 2 w : jq xk j < 1 : Qn k¼1 k n¼1 In particular qk ¼ 1 for all k, then the sequence space cesðp; qÞ reduces to cesp . We call an operator T 2 LðE; FÞ is of s-type cesðp; qÞ if 1 X n¼1

Definition 2.7 ([14,17]). Let L be the class of all bounded linear operators between arbitrary Banach spaces and LðE; FÞ be the set of all such operators from E to F. A sub collection M of L is said Tto be an ideal if each component MðE; FÞ ¼ M LðE; FÞ satisfies the following conditions: (OI1) if x0 2 E0; y 2 F then x0 y 2 MðE; F Þ (OI2) if S; T 2 MðE; F Þ then S þ T 2 MðE; F Þ (OI3) if S 2 MðE; F Þ; T 2 LðE0 ; EÞ and R 2 LðF ; F 0 Þ then RST 2 MðE0 ; F 0 Þ. Definition 2.8 ([14,17]). A function a : M ! Rþ is said to be a quasi-norm on the ideal M if the following conditions hold: (QON 1) if x0 2 E0; y 2 F then aðx0 yÞ ¼ kx0 kkyk (QON 2) if S; T 2 MðE; F Þ then there exists a constant C P 1 such that aðS þ T Þ 6 C ðaðSÞ þ aðT ÞÞ (QON 3) if S 2 MðE; F Þ; T 2 LðE0 ; EÞ and R 2 LðF ; F 0 Þ, then aðRST Þ 6 kRkaðSÞkT k.

Definition 2.9 ([14,17]). For every operator ideal M, the dual operator ideal denoted by M0 is defined as

!p < 1;

1 < p < 1:

We denote by AðsÞ p;q class of all s-type cesðp; qÞ operators between any two Banach spaces. Let q ¼ ðqk Þ be a bounded sequence of positive numbers such that q2k1 þ q2k 6 Mqk

for all k ¼ 1; 2; . . . ;

ð3:1Þ

where M > 0 is independent of k. Then we have. Theorem 3.1. Let q ¼ ðqk Þ be a bounded sequence of positive P 1 p numbers satisfying (3.1). Let 1 < p < 1. If 1 < 1, n¼1 Q n

then the class AðsÞ p;q is an operator ideal. Proof. In order to show AðsÞ p;q be an operator ideal, we prove the conditions (OI1) to (OI3). Let E and F be any two Banach spaces. Let x0 2 E0 ; y 2 F. Then x0 y is a rank one operator. So sn ðx0 yÞ ¼ 0, for all n P 2.

In particular if C ¼ 1 then a becomes a norm on the operator ideal M. An ideal M with a quasi-norm a, denoted by ½M; a is said to be a quasi-Banach operator ideal if each component MðE; FÞ is complete under the quasi-norm a. A quasi-normed operator ideal ½M; a is called injective if for every operator T 2 LðE; FÞ and a metric injection J 2 LðF; F0 Þ, JT 2 MðE; F0 Þ we have T 2 MðE; FÞ and aðJTÞ ¼ aðTÞ. Moreover, a quasi-normed operator ideal ½M; a is called surjective if for every operator T 2 LðE; FÞ and a metric surjection Q 2 LðE0 ; EÞ; TQ 2 MðE0 ; FÞ we have T 2 MðE; FÞ and aðTQÞ ¼ aðTÞ. Thus injectivity and surjectivity are dual concepts. For its various properties, please refer to [14,17–19].

n 1 X q sk ðTÞ Qn k¼1 k

We have 1 X n¼1

n 1 X q sk ðx0 yÞ Qn k¼1 k

!p

p 1 X 1 q1 s1 ðx0 yÞ Qn n¼1 p ! 1 X 1 p 0 < 1: ¼ ðq1 kx ykÞ Qn n¼1 ¼

Thus x0 y 2 AðsÞ p;q ðE ! FÞ; hence ðOI1Þ is proved. Let S; T 2 AðsÞ p;q ðE ! FÞ. Since s-number is nonnegative and nonincreasing n n n X X X qk sk ðT þ SÞ 6 q2k1 s2k1 ðT þ SÞ þ q2k s2k ðT þ SÞ k¼1

k¼1

k¼1

n X ðq2k1 þ q2k Þs2k1 ðT þ SÞ 6 k¼1

0

M ðE; FÞ ¼ fT 2 LðE; FÞ :

0

0

0

T 2 MðF ; E Þg;

where T0 is the dual of T and E0 and F0 are the duals of E and F respectively.

! n n X X qk sk ðTÞ þ qk sk ðSÞ : 6M k¼1

ð3:2Þ

k¼1


4

A. Maji, P.D. Srivastava

Using Minkowski inequality for 1 < p < 1, we have from (3.2) !p !1p 1 n X 1 X qk sk ðT þ SÞ Qn k¼1 n¼1 !p !1p 1 n n X 1 X 1 X 6M q sk ðTÞ þ q sk ðSÞ Qn k¼1 k Qn k¼1 k n¼1 2 !p !1p !p !1p 3 1 n 1 n X X X X 1 1 5 6 M4 q sk ðTÞ þ q sk ðSÞ Qn k¼1 k Qn k¼1 k n¼1 n¼1 < 1: Thus S þ T 2 AðsÞ p;q ðE ! FÞ; hence ðOI2Þ is proved. Let T 2 LðE0 ; EÞ; R 2 LðF; F0 Þ and S 2 AðsÞ p;q ðE ! FÞ. Using the property ðS3Þ in the Definition 2.2, we have 1 X n¼1

n 1 X q sk ðRSTÞ Qn k¼1 k

6 kRkkTk

1 X n¼1

!p !1p

n 1 X q sk ðSÞ Qn k¼1 k

!p !1p < 1:

Thus RST 2 AðsÞ p;q ðE0 ! F0 Þ and therefore ðOI3Þ is proved. Hence the class AðsÞ p;q is an operator ideal. h Remark 3.1. It is observed that the set AðsÞ p;q ðE ! FÞ of s-type cesðp; qÞ operators from E to F is a linear space. In particular if we take s-number sequence as the sequence of approximation numbers and qk ¼ 1, then the set AðaÞ p;q ðE ! FÞ coincides with the set of ces p type operators from E to F introduced by Constantin [9]. ðsÞ Proposition 3.1. For 1 < p 6 r < 1, we have AðsÞ p;q # Ar;q .

Proof. The result follows from the cesðp; qÞ # cesðr; qÞ for 1 < p 6 r < 1. h

bðsÞ p;q ðTÞ ¼

n¼1

1 Qn

n X

qk sk ðTÞ

;

Theorem 3.2. Let q ¼ ðqk Þ be a bounded sequence of positive P 1 p < 1, numbers satisfying (3.1). Let 1 < p < 1. If 1 n¼1 Q n

ðsÞ ^ðsÞ then the function b p;q is a quasi-norm on the operator ideal Ap;q , where

1 p X q

1 X

n 1 X yÞ ¼ qk sk ðx0 yÞ Q n n¼1 k¼1 p !1p 1 X q1 0 ¼ kx yk : Qn n¼1

!p !1p

Again kx0 yk= supkxk¼1 kðx0 yÞðxÞk = ðsupkxk¼1 jx0 ðxÞjÞkyk = kx0 kkyk. Therefore ^ðsÞ ðx0 yÞ ¼ kx0 kkyk: b p;q

Suppose that S; T 2 AðsÞ p;q ðE ! FÞ, then

!p !1p n 1 X þ TÞ ¼ q sk ðS þ TÞ Qn k¼1 k n¼1 2 !p !1p 1 n X X 1 6 M4 q sk ðSÞ Qn k¼1 k n¼1 !p !1p 3 1 n X 1 X 5 þ q sk ðTÞ Qn k¼1 k n¼1 ðsÞ 6 M bðsÞ p;q ðSÞ þ bp;q ðTÞ : ^ðsÞ ðS þ TÞ 6 M b ^ ðsÞ ðSÞ þ b ^ðsÞ ðTÞ . Thus b p;q p;q p;q bðsÞ p;q ðS

1 X

Finally, let S 2 AðsÞ p;q ðE ! FÞ, R 2 LðF; F0 Þ and T 2 LðE0 ; EÞ. Then !p !1p 1 n X 1 X ðsÞ bp ðRSTÞ ¼ q sk ðRSTÞ Qn k¼1 k n¼1 !p !1p 1 n X 1 X 6 kRkkTk q sk ðSÞ Qn k¼1 k n¼1 6 kRkbðsÞ p;q ðSÞkTk: Thus ^ðsÞ ðRSTÞ 6 kRkb ^ðsÞ ðSÞkTk: b p;q p;q h

Theorem 3.3. The operator ideal AðsÞ p;q is complete under the ^ ðsÞ i.e. ½AðsÞ ; b ^ðsÞ is a quasi-Banach operator ideal quasi-norm b p;q p;q p;q for 1 < p < 1.

k¼1

bðsÞ p;q ðTÞ

0 bðsÞ p;q ðx

^ðsÞ is a quasi-norm on the operator ideal AðsÞ . Hence b p;q p;q

!p !1p

where T 2 AðsÞ p;q .

^ðsÞ ðTÞ ¼ b p;q

Therefore,

inclusion

ðsÞ ðsÞ þ Let AðsÞ p;q be an operator ideal. Define bp;q : Ap;q ! R for 1 < p < 1 by

1 X

Since x0 y : E ! F is a rank one operator, sn ðx0 yÞ ¼ 0 for all n P 2.

Proof. Let 1 < p < 1. To prove AðsÞ p;q is a quasi-Banach operator ideal, it is enough to prove that each component ðsÞ ^ðsÞ AðsÞ p;q ðE ! FÞ of Ap;q is complete under the quasi-norm bp;q . We have bðsÞ p;q ðTÞ

!1p :

1

Qn n¼1

Proof. Let E and F be two Banach spaces and AðsÞ p;q ðE ! FÞ be any one of the components of AðsÞ p;q .

1 X

n 1 X ¼ q sk ðTÞ Qn k¼1 k n¼1 p !1p 1 X q1 P s1 ðTÞ Qn n¼1 p !1p 1 X q1 ¼ kTk ; Qn n¼1

!p !1p


5 ^ðsÞ ðTÞ for T 2 AðsÞ ðE ! FÞ: ) kTk 6 b p;q p;q

Let ðTm Þ be a Cauchy sequence in AðsÞ p;q ðE ! FÞ. Then 8 > 0, there exists n0 2 N such that ^ðsÞ ðTm Tl Þ < ; b p;q

8m; l P n0 :

ð3:4Þ

Now from (3.3), ^ ðsÞ kTm Tl k 6 b p ðTm Tl Þ. Using (3.4), we have ^ðsÞ ðTm Tl Þ < kTm Tl k 6 b p;q

8m; l P n0 :

Hence ðTm Þ is a Cauchy sequence in LðE; FÞ. As F is a Banach space, LðE; FÞ is also a Banach space. Therefore Tm ! T as m ! 1 in LðE; FÞ. We shall now show that Tm ! T as m ! 1 in AðsÞ p;q ðE ! FÞ. Using Lemma 2.1., we have jsn ðTl Tm Þ sn ðT Tm Þj 6 kTl Tk: On taking l ! 1, we have sn ðTl Tm Þ ! sn ðT Tm Þ:

ð3:5Þ

From (3.4), we get 1 X n¼1

n 1 X q sk ðTl Tm Þ Qn k¼1 k

!p !1p 0 such that b p;q all T 2 LðE; FÞ. Assume that E and F both are infinite dimensional Banach spaces. Then by Dvoretzky’s theorem [20] for m ¼ 1; 2; . . ., we have quotient spaces E=Nm and subspaces Mm of F which can be mapped onto lm 2 by isomorphisms Xm and Am such that kXm kkX1 k 6 2 and kAm kkA1 m m k 6 2. Conm sider Im be the identity map on l2 ; Qm be the quotient map from E onto E=Nm and Jm be the natural embedding map from Mm into F. Let an ; dn and un be approximation numbers, Kolmogorov numbers and Bernstein numbers [2] respectively. Then

Proof. Let 1 < p < 1. Suppose that T 2 AðaÞ p;q . Then 1 X n¼1

n 1 X q ak ðTÞ Qn k¼1 k

!p < 1:

From Proposition 2.1., we have !p !p 1 n 1 n X X 1 X 1 X q hk ðTÞ 6 q xk ðTÞ Qn k¼1 k Qn k¼1 k n¼1 n¼1 !p 1 n X 1 X 6 q ck ðTÞ Qn k¼1 k n¼1

6

1 X n¼1

n 1 X q ak ðTÞ Qn k¼1 k

!p

1 1 ¼ un ðIm Þ ¼ un ðAm A1 m Im X m X m Þ 1 6 kAm kun ðA1 m Im Xm ÞkXm k

:

1 ¼ kAm kun ðJm A1 m Im Xm ÞkXm k

This proves (I). (II) The proof of this part can be established following the technique used in part (I) above h We now state the dual of the operator ideals formed by different s-number sequences. AðaÞ p;q

Theorem 3.7. The operator ideal is symmetric and the operator ideal AðhÞ p;q is completely symmetric for 1 < p < 1. Proof. Since an ðT0 Þ 6 an ðTÞ and hn ðT0 Þ ¼ hn ðTÞ, for all ðaÞ 0 ðhÞ ðhÞ 0 T 2 LðE; FÞ, we have AðaÞ h p;q # ðAp;q Þ and Ap;q ¼ ðAp;q Þ . 0

ðdÞ Theorem 3.8. Let 1 < p < 1. Then AðcÞ and p;q ¼ ðAp;q Þ ðdÞ ðcÞ 0 Ap;q # ðAp;q Þ . In addition, if T belongs to the class of compact ðcÞ 0 operators, then AðdÞ p;q ¼ ðAp;q Þ .

Proof. The proof follows from the Theorem 2.2. Theorem 3.9. Let ðxÞ 0 AðyÞ p;q ¼ ðAp;q Þ .

p 1p

1 < p < 1.

Then

h

ðyÞ AðxÞ p;q ¼ ðAp;q Þ

Proof. The proof follows from the Theorem 2.3.

0

and

h

4. Small operator ideal This section deals with the notion of small ideal of operators. In [20], Pietsch proved that the ideal SðaÞ of approximation p type lp operators is small for 0 < p < 1. Definition 4.1. [20] An operator ideal M is said to be small if MðE; FÞ ¼ LðE; FÞ implies that at least one of the Banach spaces E and F is of finite dimension. Then we have. Theorem 4.1. The quasi-Banach ideal AðaÞ p;q of approximation type cesðp; qÞ operators is small for 1 < p < 1.

1 6 kAm kdn ðJm A1 m Im Xm ÞkXm k 1 ¼ kAm kdn ðJm A1 m Im Xm Qm ÞkXm k 1 for n ¼ 1;2;...;m: 6 kAm kan ðJm A1 m Im Xm Qm ÞkXm k

Now n n X X 1 qk 6 qk kAm kak ðJm A1 m Im Xm Qm ÞkXm k k¼1

k¼1

! n n 1 X 1 X 1 q 6 kAm k q ak ðJm Am Im Xm Qm Þ kX1 ) m k Qn k¼1 k Qn k¼1 k !p n p 1 X ) 1 6 kAm kkX1 q ak ðJm A1 m Im Xm Qm Þ m k Qn k¼1 k !1p !p !1p m m n X X 1 X 1 Therefore ð1Þ 6 kAm kkX1 k q a ðJ A I X Q Þ k m m m m m m Qn k¼1 k n¼1 n¼1 !p !1p m n 1 1 1 X 1 X 1 ) mp 6 kAm kkX1 k q a ðJ A I X Q Þ k m m m m m m a a n¼1 Qn k¼1 k 1 1 1 ^ðaÞ ) mp 6 kAm kkX1 m kbp;q ðJm Am Im Xm Qm Þ a 1 6 CkAm kkX1 m kkJm Am Im Xm Qm k 1 6 CkAm kkX1 m kkJm Am kkIm kkXm Qm k 1 ¼ CkAm kkX1 m kkAm kkXm k

6 4C:

This is a contradiction as m is any arbitrary number. Thus E and F both cannot be infinite dimensional when AðaÞ h p;q ðE ! FÞ ¼ LðE; FÞ. This completes the proof.

Acknowledgment The authors are thankful to the referees for their valuable comments and suggestions which improved the version of the paper. We also thank to Prof. N. Tita for bringing the Ref. [11] to our notice. This work was supported by CSIR, New Delhi, Grant 09/ 081(1120)/2011-EMR-I.


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On operator ideals using weighted CesØ£ ro sequence

On operator ideals using weighted CesØ£ ro sequence

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