Map Ideal of Type the Domain of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 7.40017 8.14084" width="7.40017pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-115" d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"/></g></svg>-</span>Cesàro Matrix in the Variable Exponent <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-6.1769pt" id="M2" height="17.5763pt" version="1.1" viewBox="-0.0657574 -11.3994 23.6973 17.5763" width="23.6973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-127" d="M442 569C442 628 407 660 362 660C328 660 292 648 255 616C200 568 165 508 123 334L106 264C80 242 52 222 23 202L30 176C54 191 76 205 100 222L87 166C73 103 79 43 95 19C108 0 139 -12 169 -12C242 -12 318 38 369 106L354 129C322 93 260 49 215 49C173 49 147 81 170 187L189 274C293 344 442 452 442 569ZM380 565C380 478 267 377 196 322L216 406C254 556 282 623 332 623C364 623 380 596 380 565Z"/></g><g transform="matrix(.012,0,0,-0.012,7.207,4.134)"><path id="g50-117" d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z"/></g><g transform="matrix(.012,0,0,-0.012,11.376,4.134)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.012,0,0,-0.012,15.676,4.134)"><path id="g50-47" d="M117 -12C153 -12 178 12 178 50C178 84 153 110 118 110C84 110 59 84 59 50C59 12 84 -12 117 -12Z"/></g><g transform="matrix(.012,0,0,-0.012,18.522,4.134)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g></svg> and Its Eigenvalue Distributions

Bakery, Awad A.; Mohamed, OM Kalthum S. K.

doi:https://doi.org/10.1155/2021/5567628

Journal of Function Spaces

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Variable Exponent Function Spaces and their Applications

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Volume 2021 | Article ID 5567628 | https://doi.org/10.1155/2021/5567628

Map Ideal of Type the Domain of -Cesàro Matrix in the Variable Exponent and Its Eigenvalue Distributions

Awad A. Bakery^1,2and OM Kalthum S. K. Mohamed^1,3

Academic Editor: Ciqiang Zhuo

Received10 Feb 2021

Revised02 Apr 2021

Accepted19 Apr 2021

Published28 Apr 2021

Abstract

In this article, we define a new sequence space generated by the domain of -Cesàro matrix in Nakano sequence space. Some geometric and topological properties of this sequence space, the multiplication maps defined on it, and the eigenvalue distributions of map ideal with -numbers that belong to this sequence space have been examined.

1. Introduction

The vector spaces are contained in the variable exponent spaces . Regarding the 2nd half of the twentieth century, it used to be fulfilled that these variable exponent spaces constituted the proper framework for the mathematical components of numerous issues for which the classical Lebesgue spaces have been inadequate. The relevancy of these spaces and their homes made them a famous and environment friendly device in the remedy of a range of situations; these days, the region of spaces is a prolific subject of lookup with ramifications achieving into very numerous mathematical specialties [1]. Learning about the variable exponent Lebesgue spaces obtained in addition impetus from the mathematical description of the hydrodynamics of non-Newtonian fluids [2, 3]. Applications of non-Newtonian fluids additionally known as electrorheological vary from their use in army science to civil engineering and orthopedics. By , , , and , we suggest the spaces of each, bounded, -absolutely summable and null sequences of complex numbers . We evidence the space of all, finite rank, approximable and compact bounded linear maps from a Banach space into a Banach space by , , , and , and if , we mark , , , and , respectively (see [4, 5]). The ideal of all, finite rank, approximable and compact maps is denoted by , , , and . We designate , as 1 presents at the coordinate, with .

Lemma 1 [5]. Pick up . Assume ; then, there are maps and so that , for every .

Definition 2 [5]. A Banach space is named simple if the algebra includes one and only one nontrivial closed ideal.

Theorem 3 [5]. Let be an infinite dimensional Banach space; then,

Definition 4 [6]. A map is entitled Fredholm if , , and is closed, where mentions the complement of .

Definition 5 [7]. A subclass of is named a map ideal if every component executes the next setup: (i), if illustrates Banach space of one dimension(ii) is a linear space on (iii)Suppose , , and , then , where and are normed spaces

Faried and Bakery [8] made current the notion of prequasi ideal which is added established than the quasi ideal.

Definition 6. A function is named a prequasi norm on the map ideal if the next setting encompasses the following: (1)For each , and (2)We have so as to , for all and (3)We have for , for all (4)We have if , , and ; then,

Theorem 7 [8]. is prequasi norm on the map ideal , whenever is a quasinorm on the map ideal .

Definition 8 [9]. An -number function is a map detailed on which sort to every map a nonnegative scalar sequence overbearing that the next setting encompasses the following: (a), for every (b), for each and , (c)Ideal property: , for all , , and , where and are discretionary Banach spaces(d)For and , one has (e)Rank property: assume , then , for each (f)Norming property: or , where mirrors the unit map on the -dimensional Hilbert space In an assorted illustration of -numbers, we intimate the next setting: (1)The -th Kolmogorov number, demonstrated by , is indicated by(2)The -th approximation number, established by , is denoted by

Notations 9 [8].

The ideals and multiplication mappings possess extensive grazing of mathematics in functional analysis, namely, in the theory of fixed point, eigenvalue distributions theorem, and geometric structure of Banach spaces. A few of map ideals in the class of Banach spaces or Hilbert spaces are evident by inconsistent scalar sequence spaces. For representative the ideal of compact maps is evident by the space and , for . Pietsch [5] approved the quasi-ideals , for . He investigated that the ideals of nuclear maps and of Hilbert Schmidt maps between Hilbert spaces are explored by and , respectively. He examined that are dense in , and the algebra , where , constructed simple Banach space. Pietsch [10] approved that , for , is small. Makarov and Faried [11] examined that for each infinite dimensional Banach space , and , then . Yaying et al. [12] defined and examined the sequence space, , whose -Cesàro matrix is in , with and . They explored the quasi Banach ideal of type , for and . They establish its Schauder basis, , , and duals, and found certain matrix classes connected with this sequence space. Basarir and Kara investigated the compact mappings on some Euler -difference sequence spaces [13], some difference sequence spaces of weighted means [14], the Riesz -difference sequence space [15], the -difference sequence space derived by weighted mean [16], and the order difference sequence space of generalized weighted mean [17]. Mursaleen and Noman [18, 19] introduced the compact mappings on some difference sequence spaces. The multiplication maps on Cesàro sequence spaces with the Luxemburg norm were studied by Komal et al. [20]. İlkhan et al. [21] examined the multiplication maps on Cesàro second-order function spaces. Recently, many authors in the literature have considered some nonabsolute-type sequence spaces and introduced recent high-quality papers, for example, Mursaleen and Noman [22] defined the sequence space and of nonabsolute type and proved that the spaces and are linearly isomorphic for , is a -normed space and a -space in the cases for and , and formed the basis for the space for . In [23], they examined the , , and duals of and of nonabsolute type, for . They characterized some related matrix classes and derived the characterizations of some other classes by means of a given basic lemma. On Cesàro summable sequences, Mursaleen and Başar [24] defined some spaces of double sequences whose Cesàro transforms are bounded, convergent in Pringsheim’s sense, null in Pringsheim’s sense, both convergent in Pringsheim’s sense, and bounded, regularly convergent, and absolutely -summable, respectively, and examined some topological properties of those sequence spaces. The addicted inequality will be run down in the development [25]. If and , with , and , then

Suppose , , where is the space of all sequences of positive reals, and , with , we define a new sequence space generated by the domain of -Cesàro matrix in Nakano sequence space as where and

In case , we have

Remark 10. (1)When and , with , then is compressed to , introduced and studied by Ng and Lee [26]. Different types of Cesàro summable sequence spaces of nonabsolute type have been studied by many authors [27–31](2)If , with , is truncated to studied by Yaying et al. [12]

The goal of this paper is efficient like so in Section 2 we offer the sufficient setting on any linear space of sequences , and we mark it a private sequence space (), so as to the class constructs a map ideal. We apply this theorem on . We define a subclass of the which we will call a premodular under the functional . We explain the sufficient conditions on with definite functional to become premodular . Which implies that is a prequasi normed . In Section 3, we define a multiplication map on the prequasi normed , , and give the necessity and sufficient setup on this sequence space such that the multiplication map is bounded, approximable, invertible, Fredholm, and closed range. In Section 4, firstly, we introduce the sufficient settings (not necessary) on the premodular so that is dense in . This explains a negative answer of Rhoades [32] open problem about the linearity of -type spaces. Secondly, we introduce the conditions on so that the components of prequasi ideal are complete and closed. Thirdly, we investigate the sufficient conditions on so as is precisely confined for altered powers. We explain the setup for which the prequasi ideal is minimum. Fourthly, we describe the setting for which the Banach prequasi ideal is simple. Fifthly, we expound the sufficient setting on so as to the class of all bounded linear maps which sequence of eigenvalues in equals .

2. Linear Problem

In this section, we offer the enough setting on any linear space of sequences , and we mark it private sequence space (), so as the class creates a map ideal. We apply this setting on . We define a subclass of under the functional , which we will call a premodular . We explain the enough setup on with definite functional to become premodular , which implies that is a prequasi normed .

Definition 11. The linear space of sequences is named a , if it satisfies the following: (1), with (2) is solid, i.e., for , , and , over , then (3), while illustrates the integral part of , if

Theorem 12. If the linear sequence space is , then is a map ideal.

Proof. Assume the linear sequence space is . (i)Suppose and , with . As , with and by the linearity of , one has . Therefore, , this gives (ii)Presume and , then and . As , compared to the definition of -numbers and is a decreasing sequence, we get , with . By using the linearity of , conditions (24) and (25), one can see , so (iii)Let , , and , one has . As . By using the linearity of and condition (24), we have , then Here and after, we will denote the space of all increasing sequences of real numbers by .

Theorem 13. is , if with .

Proof. (1.i)Assume . As , one hasso . (1.ii)Suppose , , and as , we obtainHence, . Relative to (1-i) and (1-ii), we have as a linear space.
Also as with , one has Therefore, , with . (1)If , for each and . One can seeHence, . (2)Assume , where , we getso .

By using Theorem 12, we can get the next theorem.

Theorem 14. Pick up with , then is a map ideal.

Definition 15. A subclass of is named a premodular , if there is a map with the following settings: (i)When , , with , where is the zero element of (ii)If and , we have with (iii) includes for some , with (iv)For , , we get (v)The inequality, includes, for (vi)If denotes the space of all sequences with finite nonzero coordinates, then (vii)We have so that , with

Definition 16. The is named a prequasi normed , if supports the points (i)–(iii) of Definition 15. If is complete equipped with , then is named a prequasi Banach .

Theorem 17. A prequasi normed , whenever it is premodular .

Theorem 18. is a premodular , if with .

Proof. (i)Easily, and (ii)We have with , for every and (iii)One has , for each (iv)Definitely, from the proof part (24) of Theorem 13(v)Indeed, from the proof part (25) of Theorem 13, (vi)Obviously, (vii)We have with , for each and , if

By following Theorems 17 and 18, we determine the next theorem.

Theorem 19. The space is a prequasi normed , if with .

3. Multiplication Maps on

In this section, we define a multiplication map on the prequasi normed and investigate the necessity and sufficient setup on so as the multiplication map is bounded, invertible, approximable, Fredholm, and closed range map.

Definition 20. If and is a prequasi normed , the map is called a multiplication map on , when , with . The multiplication map is named created by , if .

Theorem 21. Pick up and with , then , if and only if .

Proof. Suppose the settings are confirmed. Let . Hence, there is so as to , with . For , one has Therefore, .
On the contrary, let and . Hence, for all , there are so as . We have Hence, . So .

Theorem 22. Assume and be a prequasi normed . Then, , for every and with , if and only if is an isometry.

Proof. Let the sufficient condition be verified. One has with . So is an isometry.
Let the necessity condition be satisfied and , for some . We get Also, when , it is easy to show that , which is an inconsistency for the two cases. Therefore, , for all .
By , we will denote the space of all sets with finite number of elements.

Theorem 23. Raise up and with . Then, , if and only if .

Proof. Let , so . Suppose . Therefore, we have so as the set , if . Hence, is an infinite set in . Since with . Therefore, , which cannot have a convergent subsequence under . Hence, , which implies ; this gives an inconsistency. So . On the other hand, let . Hence, for all , one has . Hence, for each , we have . So . Define , for all , by It is clear that as , for all . From with , one can see Hence, . This gives that is a limit of finite rank maps. Therefore, .

Theorem 24. Pick up and with . Then, , if and only if .

Proof. Obviously, since .

Corollary 25. If with , then .

Proof. As is created the multiplication map on , and .

Theorem 26. If is a prequasi Banach and , then there are and so as , with , if and only if is closed.

Proof. Assume the sufficient setup is confirmed. Hence, there is so as , with . To show that is closed, if is a limit point of , we have , with so that . Obviously, the sequence is a Cauchy sequence. As with , one has where This implies that is a Cauchy sequence in . As is complete, there is so that . Since , we have . But . So . Hence, . Therefore, is closed. Next, suppose the necessity setup is satisfied. So there is so as , with . If , then for , one has This gives an inconsistency. Therefore, , we have , with . This proves the theorem.

Theorem 27. Pick up and be a prequasi Banach . Then, there is and so that , with , if and only if is invertible.

Proof. Let the setup be true. Assume with . By using Theorem 21, the maps and are bounded linear. We have . Therefore, . Next, let be invertible. So . Hence, is closed. Therefore, by Theorem 26, there is so that , for each . We have , if , with ; this gives which is an inconsistency, as is trivial. Therefore, , with . As , from Theorem 21, there is so that , with . Hence, one has , with .

Theorem 28. Raise up to be a prequasi Banach and . Then, is a Fredholm map, if and only if (i) is finite and (ii) , with .

Proof. Assume the sufficient condition is satisfied. Let be infinite; hence, , with . Since s are linearly independent, this gives ; this implies an inconsistency. Hence, must be finite. The condition (ii) comes from Theorem 26. Next, let the setup (i) and (ii) be confirmed. From Theorem 26, the setup (ii) implies that is closed. The setting (i) gives that and . This implies that is Fredholm.

4. Prequasi Ideal

In this section, firstly, we introduce the sufficient setting (not necessary) on such that is dense in . This investigates a negative answer of Rhoades [32] open problem about the linearity of -type spaces. Secondly, for which conditions on are complete and closed? Thirdly, we give the sufficient setup on such that is strictly contained for different powers. We explain the settings in order that is minimum. Fourthly, we explain the conditions so that the Banach prequasi ideal is simple. Fifthly, we give the sufficient conditions on such that the space of all bounded linear maps which sequence of eigenvalues in equals .

4.1. Finite Rank Prequasi Ideal

Theorem 29. , whenever with . But the converse is not necessarily true.

Proof. To show that , as , with and is a linear space, let , one has . To show that , as and , one can see . For , we have . As , suppose , then there is with , for some , where . As is decreasing, one has Therefore, there is so that rank and since , we have Therefore, one has As , and by using inequalities (5) and (24)–(27), one has Conversely, we give a counterexample as but is not verified. This confirms the proof.

4.2. Banach and Closed Prequasi Ideal

Theorem 30. The function is a prequasi norm on , where , for all , if is a premodular .

Proof. Let be a premodular ; hence, satisfies the following conditions: (1)For all , and , if and only if , for all if and only if (2)There is such that , for all and (3)There is such that for all , we have(4)There is such that if , , and , then

Theorem 31. Pick up with , then is a prequasi Banach ideal, where .

Proof. As is a premodular , hence from Theorem 30, is a prequasi norm on . Suppose is a Cauchy sequence in . As , one has so is a Cauchy sequence in . Since is a Banach space, there is with . Since , for all , from Definition 15 parts (ii), (iii), and (v), one can see Hence, , so .

Theorem 32. Suppose , be normed spaces and with , then is a prequasi closed ideal, where .

Proof. As is a premodular , from Theorem 30, is a prequasi norm on . Assume , for each and . As , we have Hence, is a convergent sequence in . Since , for every , by using Definition 15 parts (ii), (iii), and (v), one can see We get , so .

4.3. Minimum Prequasi Ideal

Theorem 33. For any infinite dimensional Banach spaces , and , with , for all , then

Proof. Suppose , then . One has Then, . After, if we choose so as , we have such that So and . Clearly, . Next, if we take such that . We have so that . This confirms the proof.

Theorem 34. Pick up any infinite dimensional Banach spaces , and with ; hence, is minimum.

Proof. Assume the setup is confirmed. So , where , is a prequasi Banach ideal. Let ; hence, there is so as , for each . Then, by Dvoretzky’s theorem [33] with , one has quotient spaces and subspaces of which can be mapped onto by isomorphisms and with and . If is the identity map on , is the quotient map from onto and is the natural embedding map from into . Assume be the Bernstein numbers [34]; hence, for . We have Hence, for some , one has We have an inconsistency, as is an arbitrary. Then, and both cannot be infinite dimensional when . This completes the proof.

Theorem 35. Upon any infinite dimensional Banach spaces , and with , then is minimum.

4.4. Simple Banach Prequasi Ideal

Theorem 36. Presume , be infinite dimensional Banach spaces. Let and with , with , then

Proof. For and . From Lemma 1, one has and with . Therefore, for each , we have This defies Theorem 33. Then, , which affects the proof.

Corollary 37. Upon any infinite dimensional Banach spaces and , if and with , for every , then

Proof. Easily, as .

Theorem 38. Raise up with , and then, be simple.

Proof. Let the closed ideal include a map . From Lemma 1, one has with . This gives that . Accordingly, . Hence, is a simple Banach space.

4.5. Eigenvalues of -Type Mappings

Notation 39.

Theorem 40. Pick up any infinite dimensional Banach spaces and . Suppose with , then

Proof. Let ; hence, and , for all . We have , with , so , with . Therefore, , so .
Secondly, let . Therefore, . Hence, we have So . Assume exists, for every . Therefore, exists and bounded, for every . So exists and bounded. From the prequasi operator ideal of , we obtain But . Therefore, , for every . This gives . This provides the proof.

5. Conclusion

In this article, we explain some topological and geometric structure, of the multiplication maps defined on , of the class , and of the class . This article has a number of advantages for researchers such as studying the fixed points of any contraction maps on this prequasi normed sequence space which is a generalization of the quasi normed sequence spaces, a new general space of solutions for many difference equations, the spectrum of any bounded linear operators between any two Banach spaces with -numbers in this sequence space and noting that the closed map ideals are certain to play an effective function in the principle of Banach lattices. We open the way for many authors to generalize the results by a sequence and generate of nonabsolute type.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia, under grant no. UJ-20-078-DR. The authors, therefore, acknowledge with thanks the university’s technical and financial support.

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Copyright © 2021 Awad A. Bakery and OM Kalthum S. K. Mohamed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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