The Spectrum of Mapping Ideals of Type Variable Exponent Function Space of Complex Variables with Some Applications

The topological and geometric behaviors of the variable exponent formal power series space, as well as the prequasi-ideal construction by s-numbers and this function space of complex variables, are investigated in this article. Upper bounds for s -numbers of infinite series of the weighted nth power forward and backward shift operator on this function space are being investigated, with applications to some entire functions.


Introduction
Operator ideal theory has various applications in the geometry of Banach spaces, xed point theory, spectral theory, and other areas of mathematics, among other areas of knowledge. Throughout the article, we will adhere to the etymological conventions listed below. If any other sources are used, we will make a note of them.
1.1. Conventions 1.1. ℕ = f0, 1, 2, ⋯g:ℂ: complex number space ℝ ℕ : the space of all real sequences ℓ ∞ : the space of bounded real sequences ℓ r : the space of r-absolutely summable real sequences c 0 : the space of null real sequences e l = ð0, 0, ⋯, 1, 0, 0, ⋯Þ, as 1 lies at the l th coordinate, for all l ∈ ℕ F: the space of each sequence with finite nonzero coordinates card ðGÞ: the number of elements of the set G mi↗: the space of all monotonic increasing sequences of positive reals L: the ideal of all bounded linear operators between any arbitrary Banach spaces F: the ideal of finite rank operators between any arbitrary Banach spaces Λ: the ideal of approximable operators between any arbitrary Banach spaces L c : the ideal of compact operators between any arbitrary Banach spaces LðX, YÞ: the space of all bounded linear operators from a Banach space X into a Banach space Y LðXÞ : the space of all bounded linear operators from a Banach space X into itself FðX, YÞ: the space of finite rank operators from a Banach space X into a Banach space Y FðXÞ: the space of finite rank operators from a Banach space X into itself ΛðX, YÞ : the space of approximable operators from a Banach space X into a Banach space Y ΛðXÞ: the space of approximable operators from a Banach space X into itself L c ðX, YÞ: the space of compact operators from a Banach space X into a Banach space Y L c ðXÞ: the space of compact operators from a Banach space X into itself ðs a ðGÞÞ a∈ℕ : the sequence of s-numbers of the bounded linear operator G ðα a ðGÞÞ a∈ℕ : the sequence of approximation numbers of the bounded linear operator G ðs a ðGÞÞ a∈ℕ : the sequence of Kolmogorov numbers of the bounded linear operator G S v : the operator ideals formed by the sequence of s -numbers in any sequence space V S app V : the operator ideals formed by the sequence of approximation numbers in any sequence space V S Kol V : the operator ideals formed by the sequence of Kolmogorov numbers in any sequence space V 1.2. Notations 1.2 (see [1] where Several operator ideals in the class of Banach or Hilbert spaces are defined by sequences of real numbers. L c , for example, is produced by ðd a ðGÞÞ a∈ℕ and c 0 . Pietsch [2] looked into the quasi-ideals S app ℓ t , for 0 < t < ∞. He demonstrated how ℓ 2 and ℓ 1 yield the ideals of Hilbert Schmidt operators and nuclear operators between Hilbert spaces, respectively. In addition, he proved that F = S ℓ t , for 1 ≤ t < ∞, and S ℓ t is a simple Banach space. Pietsch [3] explained that S ℓ t , where 0 < t < ∞, is small. Makarov and Faried [4] showed that for any Banach spaces X and Y with dim ðXÞ = dim ðYÞ = ∞, then for every r > t > 0, one has S app ℓ t ðX, Y Þ⊂ ≠ S app ℓ r ðX, YÞ⊂ ≠ LðX, YÞ. The concept of prequasi-ideal was developed by Faried and Bakery [5], who elaborated on the concept of quasi-ideal. They investigated some geometric and topological properties of the spaces S cesðtÞ and S ℓ M . According to the spectral decomposition theorem [2], for A ∈ L c ðHÞ, where H is a Hilbert space, one has AðyÞ = ∑ ∞ a=0 α a ≺ y, r a ≻ w a , where fr a g and fw a g are orthonormal families in H. Suppose ðt a Þ a∈ℕ ∈ ℝ ℕ be decreasing and D : ðη a Þ ⟶ ðt a η a Þ be the diagonal operator on ℓ p with p ≥ 1. Therefore, s a ðDÞ = t a . Shields [6] investigated an indication to the weighted shift operators as formal power series in unilateral shifts and formal Laurent series in bilateral shifts. Hedayatian [7] offered the space of formal power series with power r, H r ððbaÞÞ, where ððbaÞÞ is a sequence of positive numbers with b 0 = 1 and r > 0. By the space H p ððbaÞÞ, he meant that the set of all formal power series λ n T ð Þz n converges for any z ∈ ℂ and T − λ l T ð ÞI k k= 0, for every l ∈ ℕ : a ∈ ℕg is an orthonormal basis. Faried et al. [9] introduced the upper bounds for s-numbers of infinite series of the weighted nth power forward shift operator on H r ððbaÞÞ, for 1 ≤ r < ∞, with some applications to some entire functions. The paper is arranged as follows. In Section 3, we offer the definition of the space H pð:Þ with definite function ρ. We introduce the sufficient conditions on H pð:Þ to generate premodular special space of formal power series. This gives that H pð:Þ is a prequasinormed space. In Section 4, firstly, we give the sufficient conditions on H pð:Þ such that the class S H pð:Þ generates an operator ideal. Secondly, we explain enough settings (not necessary) on ðH pð:Þ Þ ρ , so that F = S
Definition 4 (see [2]). A Banach space Y is said to be simple if LðYÞ has one and only one nontrivial closed ideal.

Journal of Function Spaces
Theorem 8 (see [5]). Suppose g is a quasinorm on the ideal U, then g is a prequasinorm on the ideal U.
By F, we explain the space of finite formal power series, i.e, for f ∈ F, one has l ∈ ℕ with f ðzÞ = ∑ l n=0 f _ n z n .
To show that ðH pð:Þ Þ ρ is a premodular Banach (ssfps), we suppose f ðiÞ to be a Cauchy sequence in ðH pð:Þ Þ ρ , then for every ε ∈ ð0, 1Þ, there is i 0 ∈ N such that for all i, j ≥ i 0 , one gets For i, j ≥ i 0 and v ∈ ℕ, we have So Hence, f ð0Þ ∈ H pð:Þ . Then, the space ðH pð:Þ Þ ρ is a premodular Banach (ssfps). In view of Theorems 15 and 20, we conclude the following theorem.

Properties of Operator Ideal
Throughout this section, some geometric and topological properties of the prequasi-ideals formed by s-numbers and ðH pð:Þ Þ ρ are presented.  [10]). In view of Theorems 13 and 20, we conclude the next theorem.

Banach and Closed Prequasi-Ideal.
In this part, enough settings on ðH pð:Þ Þ ρ so that the prequasioperator ideal S H ρ is Banach and closed are investigated.
According to Theorem 9, we introduce the following properties of the s-type ðH pð:Þ Þ ρ .
Theorem 29. Let X and Y be Banach spaces with dim ðXÞ = dim ðYÞ = ∞, and ðp v Þ, ðq v Þ ∈ mi↗∩ ℓ ∞ with p 0 > 0 and Proof. Assume T ∈ SðH pð:Þ Þ ρ ðX, YÞ. Therefore, f s ∈ ðH pð⋅Þ Þ ρ and f s ðzÞ = ∑ ∞ v=0 s v ðTÞz v converges for any z ∈ ℂ. Then,  [15] with r ∈ ℕ, there are quotient spaces X/λ r and subspaces η r of Y that operated onto ℓ r 2 by isomorphisms D r and B r with kD r kkD −1 r k ≤ 2 and kB r kkB −1 r k ≤ 2. Suppose I r be the identity operator on ℓ r 2 , ζ r be the quotient operator from X onto X/λ r and J r be the natural embedding operator from η r into Y. Let h a be the Bernstein numbers [ for 0 ≤ j ≤ r. Then for l ≥ 1, one has As r ⟶ ∞, we get ∑ ∞ j=0 1/p j < ∞. Since ∑ ∞ j=0 a/p j ≥ 1/ sup p j ∑ ∞ j=0 1 = ∞. Hence, the space S app ðH pð⋅Þ Þ ρ is small.
By the same manner, we can easily conclude the next theorem.

Simple Prequasi-Ideal.
In this part, we offer enough settings on ðH pð⋅Þ Þ ρ so that the space S ðH pð⋅Þ Þ ρ is simple.