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Awad A. Bakery, OM Kalthum S. K. Mohamed, "Map Ideal of Type the Domain of -Cesàro Matrix in the Variable Exponent and Its Eigenvalue Distributions", Journal of Function Spaces, vol. 2021, Article ID 5567628, 12 pages, 2021. https://doi.org/10.1155/2021/5567628
Map Ideal of Type the Domain of -Cesàro Matrix in the Variable Exponent and Its Eigenvalue Distributions
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.
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 . 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 . Pick up . Assume ; then, there are maps and so that , for every .
Definition 2 . A Banach space is named simple if the algebra includes one and only one nontrivial closed ideal.
Theorem 3 . Let be an infinite dimensional Banach space; then,
Definition 4 . A map is entitled Fredholm if , , and is closed, where mentions the complement of .
Definition 5 . 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  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 . is prequasi norm on the map ideal , whenever is a quasinorm on the map ideal .
Definition 8 . 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 .
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  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  approved that , for , is small. Makarov and Faried  examined that for each infinite dimensional Banach space , and , then . Yaying et al.  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 , some difference sequence spaces of weighted means , the Riesz -difference sequence space , the -difference sequence space derived by weighted mean , and the order difference sequence space of generalized weighted mean . 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. . İlkhan et al.  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  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 , 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  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 . 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 . 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. 
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  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
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
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