#### Abstract

Suppose be sequence of positive reals. By , we represent the space of all formal power series equipped with , for some Various topological and geometric behavior of and the prequasi ideal constructs by -numbers and have been considered. The upper bounds for -numbers of infinite series of the weighted -th power forward shift operator on with applications to some entire functions are granted. Moreover, we investigate an extrapolation of Caristi’s fixed point theorem in

#### 1. Introduction

While a statement of fixed Lebesgue spaces, variable exponent Lebesgue spaces go back many years, and in successive centuries, variable Lebesgue and Sobolev spaces have been systematically examined. As then, many variable exponent real function spaces and complex function spaces have presented, for example, Hardy spaces, Besov spaces, Bessel potential spaces, Trieble-Lizorkin spaces, Morrey spaces, Herz-Morrey spaces, Fock spaces, Bergman spaces, and Herz spaces with variable exponents. For three centuries, variable exponent function spaces have been widely applied in approximation theory, image processing, and differential equations. The learn about of the variable exponent Lebesgue spaces obtained in addition impetus from the mathematical description of the hydrodynamics of non-Newtonian fluids [1, 2]. Applications of non-Newtonian fluids moreover regarded as electrorheological, vary from their use in army science to civil engineering and orthopedics. We will use the next conventions during the article; if others are used, we will state them.

*Conventions 1. *:The space of all complex numbers

:The space of all sequences of real numbers

:The space of bounded sequences of real numbers

:The space of -absolutely summable sequences of real numbers

:The space of null sequences of real numbers

:As 1 lies at the coordinate, for all

:The space of each sequences with finite nonzero coordinates

:The number of elements of G

:The space of all monotonic increasing sequences of positive reals

:The ideal of all bounded linear mappings between any arbitrary Banach spaces

:The ideal of finite rank mappings between any arbitrary Banach spaces

:The ideal of approximable mappings between any arbitrary Banach spaces

:The ideal of compact mappings between any arbitrary Banach spaces

:The space of all bounded linear mappings from a Banach space into a Banach space

:The space of all bounded linear mappings from a Banach space into itself

:The space of finite rank mappings from a Banach space into a Banach space

:The space of finite rank mappings from a Banach space into itself

:The space of approximable mappings from a Banach space into a Banach space

:The space of approximable mappings from a Banach space into itself

:The space of compact mappings from a Banach space into a Banach space

:The space of compact mappings from a Banach space into itself

:The sequence of -numbers of the bounded linear operator

:The sequence of approximation numbers of the bounded linear operator

:The sequence of Kolmogorov numbers of the bounded linear operator

:The operator ideals constructed by the sequence of -numbers in any sequence space

:The operator ideals constructed by the sequence of approximation numbers in any sequence space

:The operator ideals constructed by the sequence of Kolmogorov numbers in any sequence space .

The theory of operator ideals has many activities in fixed point theorems, eigenvalue distributions, geometry of Banach spaces, spectral theorems, etc. Some of operator ideals in the class of Banach spaces or Hilbert spaces are generated by sequence of real numbers. For example, is constructed by and . Pietsch [3] examined the quasi ideals for . He proved that the ideals of Hilbert Schmidt operators and of nuclear operators between Hilbert spaces are generated by and , respectively. Also, he examined that , for , and is simple Banach space. Pietsch [4] showed that , where , is small. Makarov and Faried [5] investigated that for any Banach spaces and with They have for all that . Faried and Bakery [6] generalized the concept of quasi ideal by introducing the concept of prequasi ideal. They introduced some geometric and topological structure of the spaces and . Yaying et al. [7] suggested the sequence space, , with its -Cesàro matrix in , with and . They examined the quasi Banach ideal of type , for and . They found its Schauder basis, , , and duals and committed to certain matrix classes linked to this sequence space. Mursaleen and Noman [8, 9] investigated the compact operators on some difference sequence spaces. The multiplication maps on Cesàro sequence spaces with the Luxemburg norm explored by Komal et al. [10]. lkhan et al. [11] affected the multiplication maps on Cesàro second order function spaces. Recently, many authors in the literature have investigated some nonabsolute kind sequence spaces and brought recent splendid papers; for example, Mursaleen and Noman [12] defined the sequence spaces 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 modeled the basis for the space for . In [13], they considered the , , and duals of and of nonabsolute type, for . They were given a picture of some related matrix classes and shown the characterizations of some other classes by means of a given basic lemma. On Cesàro summable sequences, Mursaleen and Basar [14] defined some spaces of double sequences whose Cesàro transforms are bounded, convergent in the Pringsheim’s sense, null in the Pringsheim’s sense, both convergent in the Pringsheim’s sense and bounded, regularly convergent, and absolutely -summable, respectively, and investigated some topological properties of those sequence spaces. The Banach fixed point theorem [15] gave many mathematicians the way to examine many generalizations for the contraction operators defined on the space or on the space itself. Kannan [16] examined an example of a class of operators with the identical fixed point actions as contractions but fails to be continuous. Ghoncheh [17] was the only one who investigated Kannan operators in modular vector spaces. He showed that the existence of a fixed point of Kannan mapping in complete modular spaces that have Fatou property. Bakery and Mohamed [18] explored the concept of the prequasi norm on Nakano sequence space so as to its variable exponent in . They investigated the sufficient conditions on it equipped with the definite prequasi norm to generate prequasi Banach and closed space and approved the Fatou property of distinct prequasi norms on it. Moreover, they showed that the existence of a fixed point of Kannan prequasi norm contraction maps on it and on the prequasi Banach operator ideal generated by -numbers which belong to this sequence space. According to the spectral decomposition theorem [3], for , where is a Hilbert space, then , where and are orthonormal families in . Assume be decreasing and be the diagonal operator on with . Hence, . Shields [19] gave an illustration to the weighted shift operators as formal power series in unilateral shifts and formal Laurent series in bilateral shifts. Hedayatian [20] explored the space of formal power series with power , , where is a sequence of positive numbers with and . By the space , he thought that the set of all formal power series with He investigated cyclic vectors for the forward shift operator and super cyclic vectors for the backward shift operator on the space . However, Emamirad and Heshmati [21], studied the idea of functions evident on the Bargmann space by with where is an orthonormal basis. Faried et al. [22] examined the upper bounds for -numbers of infinite series of the weighted -th power forward shift operator on , for , with some applications to some entire functions.

The aim of this paper is organized as follows: In Section 3, we introduce the definition of the space under the function . We offer the enough setup on to become premodular special space of formal power series which implies that is a prequasi normed space. In Section 4, firstly, the operator ideals generated by -numbers and so as to constructs an operator ideal are presented. Secondly, we offer the sufficient conditions (not necessary) on , such that is dense in . This presents the nonlinearity of -type spaces which offers an answer of Rhoades [23] open problem. Thirdly, we examine the setup on so that the elements of prequasi ideal are complete and closed. Fourthly, we study the enough setup on such that is strictly contained for distinct powers. We investigate the smallness of . Fifthly, we explain the simpleness of . Sixthly, we offer the enough conditions on so as to the class with its sequence of eigenvalues in is strictly contained in . In Section 5, we evaluate the upper bounds for -numbers of infinite series of the weighted -th power forward shift operator on with applications to some entire functions. Finally, in Section 6, we appraise a generalization of Caristi’s fixed point theorem in

#### 2. Definitions and Preliminaries

*Definition 2 [24]. *A map is named an -number, if the sequence , for every , verifies the next conditions:
(a)ssume , then(b), for all , , (c)The inequality satisfies, if , , and , where and are any two Banach spaces(d)If and , then (e)Assume then , whenever (f)Suppose denotes the unit map on the -dimensional Hilbert space , then or We introduce a few examples of -numbers as indicated:
(i)The -th approximation number, , where(ii)The -th Kolmogorov number, , where

*Remark 3 [24]. *Assume , where be a Hilbert space, then all the *-*numbers equal the eigenvalues of , where .

Lemma 4 [3]. *Suppose and , then there are maps and with for all .*

*Definition 5 [3]. *A Banach space is called simple if has a unique nontrivial closed ideal.

Theorem 6 [3]. *Assume be a Banach space with , one has
*

*Definition 7 [3]. *A class is called an operator ideal if each element verifies the next conditions:
(i) is linear space on (ii)Suppose , , and then,

*Definition 8 [6]. *A map is named a prequasi norm on the ideal if it verifies the next conditions:
(1)Assume , , and (2)We have with , for each and (3)We find with , for every (4)We obtain so that if , , and then , where and are normed spaces

Theorem 9 [6]. *If is a quasi norm on the ideal , then is a prequasi norm on the ideal .*

Theorem 10 [25]. *If -type Assume be an operator ideal, one has
*(1)* type *(2)*Suppose -type and -type , then -type *(3)*Assume and -type , then -type *(4)*The sequence space is solid; i.e., if -type and , for all and , then -type *

Lemma 11 [26]. *Assume be a bounded family of . Then
**The next inequality in [27] will be used in the sequel. For , and , one has
**where and .*

#### 3. Main Results

##### 3.1. The Space of Functions

In this section, we current the definition of the space equipped with the function . We introduce the sufficient setup on to form premodular special space of formal power series (ssfps). This gives that is a prequasi normed (ssfps).

Assume , we define the following space: where

When , we have

*Definition 12. *The linear space of formal power series,
is named a (ssfps) when:
(1), for every , where (2)Suppose and , for all , then (3)Let , then , where and indicates the integral part of By , we indicate the space of finite formal power series, i.e, for ; then, there is so that

*Definition 13. *A subspace of the (ssfps) is called a premodular (ssfps), if there is a map satisfies the following setup:
(i)Assume , one has and , where is the zero function of (ii)If and , one has with (iii)If , one has so as to (iv)Suppose , for every , then (v)One has such that (vi)We have with , where Remark that the continuity of at follows from part (ii). The part (1) in Definition 12 and part (vi) in Definition 13 explain that is a Schauder basis of .

The (ssfps) is named a prequasi normed (ssfps) if satisfies the parts (i)–(iii) of Definition 13, and if the space is complete equipped with , then is named a prequasi Banach (ssfps).

Theorem 14. *Every premodular (ssfps) is a prequasi normed (ssfps).*

Theorem 15. *If with , then is a premodular Banach (ssfps).*

*Proof. *(1-i) Assume . Hence, and converge for any . One has converges for any . As , one obtains
then .

(1-ii) Suppose and . Hence, converges for any . One has converges for any . As , one can see
So . Hence, from parts (1-i) and (1-ii), one gets that the space is linear. Also , for every , where and

(2) Suppose , for every and . Hence, converges for any . We have
Therefore, converges for any and . So, .

(3) If and with . Hence, converges for any and . We obtain
Therefore, converges for any and . Then, .
(i)Definitely, for all , then and (ii)One has , for every and , for so thatfor all .
(iii)One gets some satisfy the inequalityfor every .
(iv)Definitely from the proof part (2)(v)From the proof part (3), one has (vi)Definitely (vii)We have with so that , for all and , when So, the space is premodular (ssfps). To prove that is a premodular Banach (ssfps), let is a Cauchy sequence in ; hence, for all , one has so that for every , we have
For , and , one gets
Therefore, is a Cauchy sequence in , for constant ; hence, , for constant . So, , for all . Finally, to investigate that , one has
Therefore, . So, the space is a premodular Banach (ssfps).

By using Theorem 14, we offer the next theorem.☐

Theorem 16. *Assume with , then the space be a prequasi Banach (ssfps), where
*

Theorem 17. *If with , then the space is a prequasi closed (ssfps), where
*

*Proof. *From Theorem 16, the space is a prequasi normed (ssfps). To prove that is a prequasi closed (ssfps), suppose and ; then, for each , one has so that for every , we have
Therefore, for and , one obtains Hence, is a convergent sequence in , for constant . Hence, , for constant . Finally, to show that , one has
so . This explains that is a prequasi closed (ssfps).☐

#### 4. Properties of Operator Ideal

In this section, we offer some geometric and topological structure of the operator ideals generated by -numbers and .

##### 4.1. Prequasi Ideal

*Notations 18. *

Theorem 19. * is an operator ideal, whenever is a (ssfps).*

*Proof. *To prove is an operator ideal:
(i)Assume and for all , as , for every and is a linear space, then . Henceconverges for any . So and hence which gives .
(ii)Suppose and . Hence so that and converge for any . From Definition 12 part (3), we have , where and converge for any . Since , by using the definition of -numbers and is a decreasing sequence, one has for every . As from Definition 12 part (2) and is a linear space, one gets , where converges for any . Hence, .
(iii)Assume , , and . Hence, , where converges for any . As , from Definition 12 parts (1) and (2), one obtains , where converges for any . This gives . ☐

Corollary 20. *Assume with , then be an operator ideal.*

##### 4.2. Ideal of Finite Rank Operators

In this section, the sufficient conditions (not necessary) on such that is dense in are presented. This investigates the nonlinearity of the -type spaces (Rhoades open problem [23]).

Theorem 21. *, where
whenever with .*

*Proof. *Evidently, , as the space is an operator ideal. Hence, we have to show that . By choosing , hence, , with converges for any . Therefore, ; let , one has so that . Since is decreasing, one obtains