Mappings of type Orlicz and generalized Cesáro sequence space
© Mohamed and Bakery; licensee Springer 2013
Received: 7 January 2013
Accepted: 5 April 2013
Published: 18 April 2013
We study the ideal of all bounded linear operators between any arbitrary Banach spaces whose sequence of approximation numbers belong to the generalized Cesáro sequence space and Orlicz sequence space , when , ; our results coincide with that known for the classical sequence space .
Keywordsapproximation numbers operator ideal generalized Cesáro sequence space Orlicz sequence space
, for all .
, for all .
, for all , and .
, for all , .
If , for all .
where is the identity operator on the Euclidean space . Example of s-numbers, we mention approximation number , Gelfand numbers , Kolmogorov numbers and Tichomirov numbers defined by: All of these numbers satisfy the following condition:
, where is a metric injection (a metric injection is a one to one operator with closed range and with norm equal one) from the space Y into a higher space for suitable index set Λ.
for all .
, where K denotes the 1-dimensional Banach space, where .
If , then for any scalars , .
Remark 1.1 Let M be an Orlicz function then for all λ with .
Also, some geometric properties of are studied by Sanhan and Suantai .
Throughout this paper, the sequence is a bounded sequence of positive real numbers, we denote where 1 appears at i th place for all . Different classes of paranormed sequence spaces have been introduced and their different properties have been investigated. See [14–18] and .
For any bounded sequence of positive numbers , we have the following well-known inequality , and for all . See .
2 Preliminary and notation
E is a linear space and , for each .
If , and , for all , then ‘i.e. E is solid’,
if , then , where denotes the integral part of .
and , where θ is the zero element of E,
there exists a constant such that for all values of and for any scalar λ,
for some numbers , we have the inequality , for all ,
if , for all then ,
for some numbers we have the inequality ,
for each there exists such that . This means the set of all finite sequences is ρ-dense in E.
for any there exists a constant such that .
It is clear that from condition (ii) that ρ is continuous at θ. The function ρ defines a metrizable topology in E endowed with this topology is denoted by .
Example 2.2 is a pre-modular special space of sequences for , with .
Example 2.3 is a pre-modular special space of sequences for , with .
3 Main results
Theorem 3.1 is an operator ideal if E is a special space of sequences (sss).
let and for all , since E is a linear space and for each , then ; for that , which implies .
- (ii)Let and then from Definition 2.1 condition (3) we get and , since , is a decreasing sequence and from the definition of approximation numbers we get
If , and , then we get and since , from Definition 2.1 conditions (1) and (2) we get , then .
Theorem 3.2 is an operator ideal, if M is an Orlicz function satisfying -condition and there exists a constant such that .
(1-i) Let , since M is non-decreasing, we get , then .
Let for each , , since M is none decreasing, then we get , then .
Let , , then . Hence, from Theorem 3.1, it follows that is an operator ideal.
Theorem 3.3 is an operator ideal, if is an increasing sequence of positive real numbers, and .
we get , from (1-i) and (1-ii) is a linear space.
- (2)Let for each , then
Hence, . Hence, from Theorem 3.1 it follows that is an operator ideal.
Theorem 3.4 Let M be an Orlicz function. Then the linear space is dense in .
Corollary 3.5 If and , we get . See .
Theorem 3.6 The linear space is dense in , if is an increasing sequence of positive real numbers with and .
Theorem 3.7 Let X be a normed space, Y a Banach space and be a pre modular special space of sequences (sss), then is complete.
Hence as such . □
Corollary 3.8 Let X be a normed space, Y a Banach space and M be an Orlicz function such that M satisfies -condition. Then M is continuous at and is complete.
Corollary 3.9 Let X be a normed space, Y a Banach space and be an increasing sequence of positive real numbers with and , then is complete.
Dedicated to Professor Hari M Srivastava.
The authors wish to thank the referees for their careful reading of the paper and for their helpful suggestions.
- Kalton NJ: Spaces of compact operators. Math. Ann. 1974, 208: 267–278. 10.1007/BF01432152MathSciNetView ArticleGoogle Scholar
- Lima Å, Oja E: Ideals of finite rank operators, intersection properties of balls, and the approximation property. Stud. Math. 1999, 133: 175–186. MR1686696 (2000c:46026)MathSciNetGoogle Scholar
- Pietsch A: Operator Ideals. North-Holland, Amsterdam; 1980. MR582655 (81j:47001)Google Scholar
- Krasnoselskii MA, Rutickii YB: Convex Functions and Orlicz Spaces. Noordhoff, Groningen; 1961.Google Scholar
- Orlicz, W: Über Raume ( L M ). Bull. Int. Acad Polon. Sci. A, 93–107 (1936)Google Scholar
- Nakano H: Concave modulars. J. Math. Soc. Jpn. 1953, 5: 29–49. 10.2969/jmsj/00510029View ArticleGoogle Scholar
- Maddox IJ: Sequence spaces defined by a modulus. Math. Proc. Camb. Philos. Soc. 1986, 100(1):161–166. 10.1017/S0305004100065968MathSciNetView ArticleGoogle Scholar
- Ruckle WH: FK spaces in which the sequence of coordinate vectors is bounded. Can. J. Math. 1973, 25: 973–978. 10.4153/CJM-1973-102-9MathSciNetView ArticleGoogle Scholar
- Tripathy BC, Chandra P: On some generalized difference paranormed sequence spaces associated with multiplier sequences defined by modulus function. Anal. Theory Appl. 2011, 27(1):21–27. 10.1007/s10496-011-0021-yMathSciNetView ArticleGoogle Scholar
- Lindenstrauss J, Tzafriri L: On Orlicz sequence spaces. Isr. J. Math. 1971, 10: 379–390. 10.1007/BF02771656MathSciNetView ArticleGoogle Scholar
- Altin Y, Et M, Tripathy BC:The sequence space on seminormed spaces. Appl. Math. Comput. 2004, 154: 423–430. 10.1016/S0096-3003(03)00722-7MathSciNetView ArticleGoogle Scholar
- Et M, Altin Y, Choudhary B, Tripathy BC: On some classes of sequences defined by sequences of Orlicz functions. Math. Inequal. Appl. 2006, 9(2):335–342.MathSciNetGoogle Scholar
- Sanhan W, Suantai S: On k -nearly uniformly convex property in generalized Cesáro sequence space. Int. J. Math. Math. Sci. 2003, 57: 3599–3607.MathSciNetView ArticleGoogle Scholar
- Rath D, Tripathy BC: Matrix maps on sequence spaces associated with sets of integers. Indian J. Pure Appl. Math. 1996, 27(2):197–206.MathSciNetGoogle Scholar
- Tripathy BC, Sen M: On generalized statistically convergent sequences. Indian J. Pure Appl. Math. 2001, 32(11):1689–1694.MathSciNetGoogle Scholar
- Tripathy BC, Hazarika B: Paranormed I -convergent sequences. Math. Slovaca 2009, 59(4):485–494. 10.2478/s12175-009-0141-4MathSciNetView ArticleGoogle Scholar
- Tripathy BC, Sen M: Characterization of some matrix classes involving paranormed sequence spaces. Tamkang J. Math. 2006, 37(2):155–162.MathSciNetGoogle Scholar
- Tripathy BC: Matrix transformations between some classes of sequences. J. Math. Anal. Appl. 1997, 206: 448–450. 10.1006/jmaa.1997.5236MathSciNetView ArticleGoogle Scholar
- Tripathy BC: On generalized difference paranormed statistically convergent sequences. Indian J. Pure Appl. Math. 2004, 35(5):655–663.MathSciNetGoogle Scholar
- Altay Band Başar F:Generalization of the sequence space derived by weighted means. J. Math. Anal. Appl. 2007, 330(1):147–185.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.