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

The vector spaces ltð:Þ are contained in the variable exponent spaces Ltð:Þ. 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 Ltð:ÞðΩÞ spaces is a prolific subject of lookup with ramifications achieving into very numerous mathematical specialties [1]. Learning about the variable exponent Lebesgue spaces Ltð:Þ obtained in addition impetus from the mathematical description of the hydrodynamics of non-Newtonian fluids [2, 3]. Applications of nonNewtonian fluids additionally known as electrorheological vary from their use in army science to civil engineering and orthopedics. By CN, l∞, lr , and c0, we suggest the spaces of each, bounded, r-absolutely summable and null sequences of complex numbers N = f0, 1, 2,⋯g. We evidence the space of all, finite rank, approximable and compact bounded linear maps from a Banach space P into a Banach space Q by BðP ,QÞ, FðP ,QÞ, AðP ,QÞ, and KðP ,QÞ, and if P =Q, we mark BðP Þ, FðP Þ, AðP Þ, and KðP Þ, respectively (see [4, 5]). The ideal of all, finite rank, approximable and compact maps is denoted by B, F, A , and K . We designate el = ð0, 0, ⋯, 1, 0, 0,⋯Þ, as 1 presents at the lth coordinate, with l ∈N.


Introduction
The vector spaces ℓ tð:Þ are contained in the variable exponent spaces L tð:Þ . 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 L tð:Þ ðΩÞ spaces is a prolific subject of lookup with ramifications achieving into very numerous mathematical specialties [1]. Learning about the variable exponent Lebesgue spaces L tð:Þ 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 C N , ℓ ∞ , ℓ r , and c 0 , we suggest the spaces of each, bounded, r-absolutely summable and null sequences of complex numbers N = f0, 1, 2, ⋯g. We evidence the space of all, finite rank, approximable and compact bounded linear maps from a Banach space P into a Banach space Q by BðP , QÞ, FðP , QÞ, AðP , QÞ, and KðP , QÞ, and if P = Q, we mark BðP Þ, FðP Þ, AðP Þ, and KðP Þ, respectively (see [4,5]). The ideal of all, finite rank, approximable and compact maps is denoted by B, F, A, and K. We designate e l = ð0, 0, ⋯, 1, 0, 0, ⋯Þ, as 1 presents at the l th coordinate, with l ∈ N. Lemma 1 [5]. Pick up U ∈ BðP , QÞ. Assume U ∉ AðP , QÞ; then, there are maps X ∈ BðP Þ and Y ∈ BðQÞ so that YUXe l = e l , for every l ∈ N.
Definition 2 [5]. A Banach space V is named simple if the algebra BðV Þ includes one and only one nontrivial closed ideal.
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 c 0 and d a ðXÞ, for X ∈ BðP , QÞ. Pietsch [5] approved the quasi-ideals He investigated that the ideals of nuclear maps and of Hilbert Schmidt maps between Hilbert spaces are explored by ℓ 1 and ℓ 2 , respectively. He examined that Fð ℓ b Þ are dense in Bðℓ b Þ, and the algebra Bðℓ b Þ, where ð1 ≤ b< ∞Þ, constructed simple Banach space. Pietsch [10] approved that B α ℓ b , for 0 < b < ∞, is small. Makarov and Faried [11] examined that for each infinite dimensional Banach space P , Q and r > b > 0, then B α ℓ b ðP , QÞÞB α ℓ r ðP , QÞUBðP , QÞ. Yaying et al. [12] defined and examined the sequence space, χ t r , whose r-Cesàro matrix is in ℓ t , with r ∈ ð0, 1 and 1 ≤ t Journal of Function Spaces ≤ ∞. They explored the quasi Banach ideal of type χ t r , for r ∈ ð0, 1 and 1 < t < ∞. They establish its Schauder basis, α − , β − , and γ − duals, and found certain matrix classes connected with this sequence space. Basarir and Kara investigated the compact mappings on some Euler BðmÞ-difference sequence spaces [13], some difference sequence spaces of weighted means [14], the Riesz BðmÞ-difference sequence space [15], the B-difference sequence space derived by weighted mean [16], and the m th 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 ℓ λ p and ℓ λ ∞ of nonabsolute type and proved that the spaces ℓ λ p and ℓ λ p are linearly isomorphic for 0 < p ≤ ∞, ℓ λ p is a p-normed space and a BK-space in the cases for 0 < p < 1 and 1 ≤ p ≤ ∞, and formed the basis for the space ℓ λ p for 1 ≤ p < ∞. In [23], they examined the α − , β − , and γ − duals of ℓ λ p and ℓ λ ∞ of nonabsolute type, for 1 ≤ p < ∞. 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 q-summable, respectively, and examined some topological properties of those sequence spaces. The addicted inequality will be run down in the development [25]. If r a ≥ 1 and x a , z a ∈ C, with a ∈ N, and ℏ = sup a r a , then Suppose r ∈ ð0, 1Þ, ðt l Þ ∈ R +N , where R +N is the space of all sequences of positive reals, and t l ≥ 1, with l ∈ N, we define a new sequence space generated by the domain of r-Cesàro matrix in Nakano sequence space as where υð f Þ = ∑ ∞ l=0 ðj∑ l z=0 r z f z j/½l + 1 r Þ t l and l, r = 1: In case ðt l Þ ∈ ℓ ∞ , we have Remark 10.
(1) When r = 1 and t l = t, with l ∈ N, then ces ðtÞ r is compressed to ces t , 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][28][29][30][31] (2) If t l = t, with l ∈ N, ces ðtÞ r is truncated to χ t r 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 V , and we mark it a private sequence space (pss), so as to the class B s V constructs a map ideal. We apply this theorem on ces ðtÞ r . We define a subclass of the pss which we will call a premodular pss under the functional υ : V ⟶ ½0,∞Þ. We explain the sufficient conditions on ces ðtÞ r with definite functional υ to become premodular pss. Which implies that ces ðtÞ r is a prequasi normed pss. In Section 3, we define a multiplication map on the prequasi normed pss, ðces ðtÞ r Þ υ , 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 pssðces

Linear Problem
In this section, we offer the enough setting on any linear space of sequences V , and we mark it private sequence space (pss), so as the class B s V creates a map ideal. We apply this setting on ces ðtÞ r . We define a subclass of pss under the functional υ : V ⟶ ½0,∞Þ, which we will call a premodular pss.
We explain the enough setup on ces ðtÞ r with definite 3 Journal of Function Spaces functional υ to become premodular pss, which implies that ces ðtÞ r is a prequasi normed pss.
Definition 11. The linear space of sequences V is named a pss, if it satisfies the following: Proof. Assume the linear sequence space V is pss.
As e b ∈ V , with b ∈ N and by the linearity of V , to the definition of s-numbers and s b ðXÞ is a decreasing sequence, we get s b ðρ 1 X 1 + ρ 2 X 2 Þ ≤ s 2½b/2 ðρ 1 X 1 + ρ 2 X 2 Þ ≤ s ½b/2 ðρ 1 X 1 Þ + s ½b/2 ðρ 2 X 2 Þ = jρ 1 js ½b/2 ðX 1 Þ + jρ 2 js ½b/2 ðX 2 Þ, with b ∈ N. By using the linearity of V , conditions (24) and (25), one can see As s b ðZYXÞ ≤ kZks b ðYÞkXk. By using the linearity of V and condition (24), we have Here and after, we will denote the space of all increasing sequences of real numbers by I.
so f + g ∈ ces ðtÞ r .
By using Theorem 12, we can get the next theorem.  By following Theorems 17 and 18, we determine the next theorem.

Multiplication Maps on ðces ðtÞ r Þ υ
In this section, we define a multiplication map on the prequasi normed pssðces ðtÞ r Þ υ and investigate the necessity and sufficient setup on ðces ðtÞ r Þ υ so as the multiplication map is bounded, invertible, approximable, Fredholm, and closed range map.

Theorem 22.
Assume ω ∈ C N and ðces ðtÞ r Þ υ be a prequasi normed pss. Then, ω b = g, for every b ∈ N and g ∈ C with jgj = 1, if and only if H ω is an isometry.
Proof. Let the sufficient condition be verified. One has with f ∈ ðces ðtÞ r Þ υ . So H ω is an isometry. Let the necessity condition be satisfied and jω b j < 1, for Also, when jω b 0 j > 1, it is easy to show that υðH ω e b 0 Þ > υ ðe b 0 Þ, which is an inconsistency for the two cases. Therefore, By F, we will denote the space of all sets with finite number of elements. Theorem 23. Raise up ω ∈ C N and ðt l Þ ∈ I ∩ ℓ ∞ with t 0 > 1.