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

Bakery, Awad A.; Mohammed, Mustafa M.

doi:https://doi.org/10.1155/2021/3526217

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Developments in Functions and Operators of Complex Variables

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Volume 2021 | Article ID 3526217 | https://doi.org/10.1155/2021/3526217

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

Awad A. Bakery^1,2and Mustafa M. Mohammed^1,3

Academic Editor: Sibel Yalçın

Received19 Jun 2021

Accepted10 Oct 2021

Published29 Oct 2021

Abstract

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

1. 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

: complex number space

: the space of all real sequences

: the space of bounded real sequences

: the space of -absolutely summable real sequences

: the space of null real sequences

, as 1 lies at the coordinate, for all

: the space of each sequence with finite nonzero coordinates

: the number of elements of the set

: the space of all monotonic increasing sequences of positive reals

: the ideal of all bounded linear operators between any arbitrary Banach spaces

: the ideal of finite rank operators between any arbitrary Banach spaces

: the ideal of approximable operators between any arbitrary Banach spaces

: the ideal of compact operators between any arbitrary Banach spaces

: the space of all bounded linear operators from a Banach space into a Banach space

: the space of all bounded linear operators from a Banach space into itself

: the space of finite rank operators from a Banach space into a Banach space

: the space of finite rank operators from a Banach space into itself

: the space of approximable operators from a Banach space into a Banach space

: the space of approximable operators from a Banach space into itself

: the space of compact operators from a Banach space into a Banach space

: the space of compact operators 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 formed by the sequence of -numbers in any sequence space

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

: the operator ideals formed by the sequence of Kolmogorov numbers in any sequence space

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. , for example, is produced by and . Pietsch [2] looked into the quasi-ideals , for . He demonstrated how and yield the ideals of Hilbert Schmidt operators and nuclear operators between Hilbert spaces, respectively. In addition, he proved that , for , and is a simple Banach space. Pietsch [3] explained that , where , is small. Makarov and Faried [4] showed that for any Banach spaces and with , then for every , one has . 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 and . According to the spectral decomposition theorem [2], for , where is a Hilbert space, one has , where and are orthonormal families in . Suppose be decreasing and be the diagonal operator on with . Therefore, . 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 , , where is a sequence of positive numbers with and . By the space , he meant that the set of all formal power series with . He studied cyclic vectors for the forward shift operator and supercyclic vectors for the backward shift operator on the space .

However, Emamirad and Heshmati [8] explored the idea of functions evident on the Bargmann space by with , where is an orthonormal basis. Faried et al. [9] introduced 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 paper is arranged as follows. In Section 3, we offer the definition of the space with definite function . We introduce the sufficient conditions on to generate premodular special space of formal power series. This gives that is a prequasinormed space. In Section 4, firstly, we give the sufficient conditions on such that the class generates an operator ideal. Secondly, we explain enough settings (not necessary) on , so that . This shows the nonlinearity of -type spaces which gives an answer of Rhoades [10] open problem. Thirdly, we investigate the conditions on such that the prequasi-ideal are Banach and closed. Fourthly, we examine the sufficient conditions on such that is strictly contained for different powers. We show the smallness of . Fifthly, we investigate the simpleness of . Sixthly, we present the enough setup on such that the class with its sequence of eigenvalues in equals . In Section 5, we estimate the upper bounds for -numbers of infinite series of the weighted th power forward and backward shift operator on with approaches to some entire functions.

2. Definitions and Preliminaries

Definition 1 (see [11]). A function is called an -number, if the sequence , for all , shows the following settings: (a)If , then (b), for every (c)The inequality holds, if and , where and are arbitrary Banach spaces(d)Suppose and , then (e)Let then , whenever (f)Assume indicates the identity operator on the -dimensional Hilbert space , then or Consider the following examples of -numbers: (i)The th approximation number, , where (ii)The th Kolmogorov number, , where

Remark 2 (see [11]). If , where be a Hilbert space, then all the -numbers equal the eigenvalues of , where .

Lemma 3 (see [2]). If and , then and with , for each .

Definition 4 (see [2]). A Banach space is said to be simple if has one and only one nontrivial closed ideal.

Theorem 5 (see [2]). If D is a Banach space with , then

Definition 6. (see [2]). A class is said to be an operator ideal if every vector shows the following settings: (i)(ii) is linear space on (iii)If and then,

Definition 7 (see [5]). A function is called a prequasinorm on the ideal if it shows the next settings: (1)For each and (2)One has with , for all and (3)One has with , for every (4)There exists so that if and then , where and are normed spaces

Theorem 8 (see [5]). Suppose is a quasinorm on the ideal , then is a prequasinorm on the ideal .

Theorem 9 (see [12]). Assume . If is an operator ideal, then we have (1)-type (2)Assume (3)Suppose and (4)The sequence space is solid. i.e., when

Lemma 10 (see [13]). If is a bounded family of . We have

Lemma 11 (see [14]). If , then

Definition 12 (see [1]). The linear space of formal power series is called a special space of formal power series (or in short (ssfps)), if it shows the following settings: (1), for all , where (2)If and (3)Suppose , then , where marks the integral part of

Theorem 13 (see [1]). If is a (ssfps), then is an operator ideal.
By , we explain the space of finite formal power series, i.e, for , one has with .

Definition 14 (see [1]). A subspace of the (ssfps) is called a premodular (ssfps), if there is a function verifies the next conditions: (i)For , we have and , where is the zero function of (ii)Suppose and , then there is with (iii)Let , then there is such that (iv)Suppose , for every , then (v)There is so that (vi)(vii)one has with , where Note that the continuity of at comes from condition (ii). Condition (1) in Definition 12 and condition (vi) in Definition 14 investigate that is a Schauder basis of .
The (ssfps) is called a prequasinormed (ssfps) if ρ shows the conditions (i)–(iii) of Definition 14, and if the space is complete under , then is called a prequasi-Banach (ssfps).

Theorem 15 (see [1]). Every premodular (ssfps) is a prequasinormed (ssfps).

Definition 16 (see [1]). Assume is a prequasinormed (ssfps). An operator is called forward shift, if , for all , where converges for every and .

Definition 17 (see [1]). Suppose is a prequasinormed (ssfps). An operator is called backward shift, if , for all , where converges for every and .

Definition 18 (see [9]). By using the power series of an entire function , the shift operator is defined as

Definition 19 (see [9]). By using the power series of an entire function , the shift operator is defined as

3. Main Results

3.1. The Space of Functions

We define in this section the space under the function and give enough conditions on it to create pre-modular (ssfps) which implies that is a prequasi-Banach (ssfps).

If , we define the new space of functions: where

If , one has

Theorem 20. Consider with , one has is a premodular Banach (ssfps).

Proof (1-i). Let . Therefore, converge for any . Then, . From , we haveso .
(1-ii) Let . Therefore, converges for any . Then, converges for any . From , we have
So . Therefore, from conditions (1-i) and (1-ii), the space is linear. To prove ,
for all , where (2)Assume , for all and . Then,converges for any One has So, and and . Hence, (3)Let and with . Then, converges for any and . One has Hence, converges for any and . Then . (i)Obviously, if , one gets and (ii)There is , for all and , for so that for all (iii)There is so that for every , (iv)Obviously from the proof part (2).(v)From the proof part (3), one has (vi)Clearly, (vii)One has with with , for each and , when . Therefore, the space is a premodular (ssfps). To show that is a premodular Banach (ssfps), we suppose to be a Cauchy sequence in , then for every , there is such that for all , one gets For and , we have So, is a Cauchy sequence in , for fixed , hence , for fixed .
Therefore, , for every . Finally, to show that , we have Hence, . Then, the space is a premodular Banach (ssfps).
In view of Theorems 15 and 20, we conclude the following theorem.

Theorem 21. If with , then the space is a prequasi-Banach (ssfps), where

Theorem 22. Suppose with , one has is a prequasiclosed (ssfps), where

Proof. According to Theorem 21, the space is a prequasinormed (ssfps). To explain that is a prequasiclosed (ssfps), let and , we have for all , there is such that for all , one gets So, for and , we have .Therefore, is a convergent sequence in , for fixed . Then, for fixed . Finally to prove that , we have this gives which shows that is a prequasiclosed (ssfps).

4. Properties of Operator Ideal

Throughout this section, some geometric and topological properties of the prequasi-ideals formed by -numbers and are presented.

4.1. Ideal of Finite Rank Operators

In this part, enough settings (not necessary) on so that are given. This explains the nonlinearity of the -type spaces (Rhoades open problem [10]).

In view of Theorems 13 and 20, we conclude the next theorem.

Theorem 23. Consider with , then is an operator ideal.

Theorem 24. If with , then , where

Proof. Clearly, , since the space is an operator ideal. Therefore, we have to show that . By letting , then, , with converges for any . So, , fix , we have with . As is decreasing, we have Therefore, we have , rank and As , then Since , then , where converges for any . Because is increasing and from the inequalities (27)–(29), we get Since but the condition is not verified which explain a negative example of the converse statement. This finishes the proof.
We can reformulate Theorem 24 as follows: if with , then every compact operators can be approximated by finite rank operators and the converse is not always true.

4.2. Banach and Closed Prequasi-Ideal

In this part, enough settings on so that the prequasioperator ideal is Banach and closed are investigated.

Theorem 25. Assume with , then the function is a prequasinorm on , where converges for any and

Proof. One has verifies the next setups: (1)Let and , for all (2)There is with , for every and (3)One has , for . Then, and converge for any . Therefore, for , one has(4)We have , let and . Then, converges for all . Then, for , one has

Theorem 26. Assume and are Banach spaces, and with , then is a prequasi-Banach operator ideal, where converges for any and

Proof. As with , one has the function is a prequasinorm on . Let be a Cauchy sequence in . Therefore, and converges for any . Suppose , then from parts (iv) and (vii) of Definition 14 and since , we have then is a Cauchy sequence in . Since the space is a Banach space, there is with and as , for every . Hence, by using Theorem 25 and the continuity of at , we have so , which implies .

Theorem 27. Suppose and are Banach spaces, and with , then is a prequasiclosed operator ideal, where converges for any and

Proof. As with , so the function is a prequasinorm on . Let , with and . Then, and converges for any . Suppose , then from parts (iv) and (vii) of Definition 14 and since , one obtains then is a convergent sequence in . Since the space is a Banach space, then there is with and as , for every , by using Theorem 25 and the continuity of ρ at θ, one has hence, , which gives .

According to Theorem 9, we introduce the following properties of the -type .

Theorem 28. For -type . The next settings are verified. (1)We have -type (2)Suppose and , then (3)One has and , then (4)The -type is solid

4.3. Small Prequasi-Banach Ideal

We introduce here some inclusion relations concerning the space for different .

Theorem 29. Let and be Banach spaces with , and with and , for all , we have

Proof. Assume . Therefore, and converges for any . Then, hence, . Next, by taking T with , one has and . Clearly, . Again, by choosing , one has and . This finishes the proof.
In this part, we examine the sufficient setting for which is small.

Theorem 30. Let and be Banach spaces with . Assume with , then is small.

Proof. Obviously, the space generates a prequasi-Banach operator ideal, with . Let . Hence, there is with , for all . According to Dvoretzky’s theorem [15] with , there are quotient spaces and subspaces of that operated onto by isomorphisms and with and . Suppose be the identity operator on , be the quotient operator from onto and J_r be the natural embedding operator from into . Let be the Bernstein numbers [16], we have for . Then for , one has As , we get . Since . Hence, the space is small.
By the same manner, we can easily conclude the next theorem.

Theorem 31. Assume and be Banach spaces with . Suppose with , then is small.

4.4. Simple Prequasi-Ideal

In this part, we offer enough settings on so that the space is simple.

Theorem 32. Let , with , for every , then

Proof. Consider and . According to Lemma 3, one has and with . For every , one obtains This defies Theorem 29.

Corollary 33. Let with , for each , then

Proof. Clearly, as .

Theorem 34. Assume with , then is simple.

Proof. Suppose and . In view of Lemma 3, we have so as to . One gets . Therefore, . This implies one and only one nontrivial closed ideal in .

4.5. Spectrum of Prequasi-Ideal

In this part, we introduce enough settings on so that the class with sequence of eigenvalues in equals .

Theorem 35. If and are Banach spaces with . Suppose with , we have

Proof. Let , then , where converges for all with , and for all . We have , with , hence , with . As a result, , then . Secondly, assume . Hence, , where converges for all with . One has Therefore, . Let exists, for all . Hence, exists and bounded, for all . Therefore, exists and bounded. By using the prequasioperator ideal of , one has Since . Hence, , for all . This gives .
This shows the proof.

5. Weighted Shift Operators on

In this section, we present the upper bounds of -numbers for infinite series of the weighted th power forward and backward shift operator on with applications to some entire functions.

Theorem 36. Assume with , then with where , for all .

Proof. Suppose the setups are verified. For . Since with , then
Therefore, with . Since . Then, there is with , for all . Hence, , one gets .
This completes the proof.

Theorem 37. Consider with , then with where , for every .

Proof. Let the given settings hold for every . Since with , then Therefore, with . Since . Then, there is with , for all . Hence, , then . This completes the proof.

Theorem 38. Let with . Suppose , then every function in is analytic on the open unit disc . Moreover, the convergence in implies the uniform convergence on compact subsets of , where , for any .

Proof. Suppose , and .Therefore, converges for every and . Hence, . We have Since with , we obtain , for all . Hence, converges for every complex value of . Assume is a compact subset of and , for all . Let converges to , we have where with and , for all . Clearly, , then . So .

Theorem 39. Assume is the forward shift operator on , with , for all . Then, where .

Proof. Let card and as , for all , where converges for every and . Hence, and .Assume is and operator on with rank defined by Since . This implies .
Define an operator by , then Hence, , where Therefore, the identity map will be , according to the definition of -numbers, we have This inequality is satisfied for all card, and one has On the other hand, let ξ be a subset of with card . Define the finite rank map by . In view of the definition of approximation numbers, we have This inequality is verified for every card and by using Lemma 10, one has This completes the proof.

Theorem 40. If is the backward shift operator on , with , for all Then, where .

Proof. Assume card and since , for every , where converges for any . Therefore, .Suppose is an operator on with rank evident by As . This implies . Define an operator , one gets Therefore, , where . Hence, the identity operator will be , in view of the definition of-numbers, one has This inequality is confirmed for all card , and we have On the other hand, suppose is a subset of with card . Define the finite rank operator by . From the definition of approximation numbers, one gets This inequality is satisfied for any card and from Lemma 10, we have This finishes the proof.

Next, the upper and lower bounds of norm have been explained.

Theorem 41. The effect of

Proof. Assume . Then, Since satisfies the triangle inequality, we get Next the upper and lower bounds of norm have been investigated.

Theorem 42. The effect of , we have

Proof. Suppose . We have As verifies the triangle inequality, one can see The following theorem indicates an upper estimation to the -numbers of .

Theorem 43. The effect of , the -numbers of this operator are presented by for all .

Proof. Let ξ be a subset of and card . By using the definition of -numbers. Define the finite rank operator . In view of the definition of approximation numbers and since satisfies the triangle inequality, we have This inequality is verified for every card , and one has This implies the proof.

The next theorem investigates an upper estimation to the -numbers of .

Theorem 44. Acting on the space , where , the -numbers of this operator satisfy for all .

Proof. Assume is a subset of and card From the definition of -numbers. Define the finite rank operator . From the definition of approximation numbers and as verifies the triangle inequality, one has This inequality is satisfied for all card , and we have This completes the proof.

The following theorems are direct consequences of Theorem 43 and Definition 18, for some entire functions, for example, the exponential and the sine functions.

Theorem 45. Let . Assume is a shift operator on , where for . The upper estimation of the s-number of is given by

Theorem 46. Let . Suppose is a shift operator on , where . The upper estimation of the s-numbers of is presented by

The following theorems are direct consequences of Theorem 44 and Definition 19, for some entire functions, for example, the exponential and the sine functions.

Theorem 47. Assume . Suppose is a shift operator on , where . The upper estimation of the -numbers of is pretended by

Theorem 48. Suppose . Assume is a shift operator on , where . The upper estimation of the is presented by

Data Availability

No data were used.

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-20-080-DR. The authors, therefore, acknowledge with thanks the University’s technical and financial support.

References

A. A. Bakery and M. H. El Dewaik, “A generalization of Caristi’s fixed point theorem in the variable exponent weighted formal power series space,” Journal of Function Spaces, vol. 2021, Article ID 9919420, 18 pages, 2021.
View at: Publisher Site | Google Scholar
A. Pietsch, Operator Ideals, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1980.
A. Pietsch, “Small ideals of operators,” Studia Mathematica, vol. 51, no. 3, pp. 265–267, 1974.
View at: Publisher Site | Google Scholar
B. M. Makarov and N. Faried, “Some properties of operator ideals constructed by s numbers (In Russian),” in Theory of Operators in Functional Spaces, pp. 206–211, Academy of Science, Siberian section, Novosibirsk, Russia, 1977.
View at: Google Scholar
N. Faried and A. A. Bakery, “Small operator ideals formed by s numbers on generalized Cesáro and Orlicz sequence spaces,” Journal of Inequalities and Applications, vol. 2018, no. 1, 2018.
View at: Publisher Site | Google Scholar
A. L. Shields, “Weighted shift operators and analytic function theory,” in Topics of Operator Theory, in: Mathematical Surveys Monographs, vol. 13, American Mathematical Society, Providence, RI, USA, 1974.
View at: Google Scholar
K. Hedayatian, “On cyclicity in the space H^p(β),” Taiwanese Journal of Mathematics, vol. 8, no. 3, pp. 429–442, 2004.
View at: Publisher Site | Google Scholar
H. Emamirad and G. S. Heshmati, “Chaotic weighted shifts in Bargmann spaces,” Journal of mathematical analysis and applications, vol. 308, no. 1, pp. 36–46, 2005.
View at: Publisher Site | Google Scholar
N. Faried, A. Morsy, and Z. A. Hassanain, “s-numbers of shift operators of formal entire functions,” Journal of Approximation Theory, vol. 176, pp. 15–22, 2013.
View at: Publisher Site | Google Scholar
B. E. Rhoades, “Operators of A − p type,” Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, vol. 59, no. 3-4, pp. 238–241, 1975.
View at: Google Scholar
A. Pietsch, Eigenvalues and S-Numbers, Cambridge University Press, New York, NY, USA, 1986.
A. A. Bakery and A. R. Abou Elmatty, “A note on Nakano generalized difference sequence space,” Advances in Difference Equations, vol. 2020, no. 1, 17 pages, 2020.
View at: Publisher Site | Google Scholar
N. Faried, Z. Abd El Kader, and A. A. Mehanna, “s-numbers of polynomials of shift operators on ℓ^p 1 ≤ p ≤ ∞,” Journal of the Egyptian Mathematical Society, vol. 1, pp. 31–37, 1993.
View at: Google Scholar
B. Altay and F. Başar, “Generalization of the sequence space ℓ(p) derived by weighted means,” Journal of Mathematical Analysis and Applications, vol. 330, no. 1, pp. 147–185, 2007.
View at: Google Scholar
A. Pietsch, Operator Ideals, VEB Deutscher Verlag derWissenschaften, Berlin, 1978.
A. Pietsch, “s-numbers of operators in Banach spaces,” Studia Mathematica, vol. 51, no. 3, pp. 201–223, 1974.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Awad A. Bakery and Mustafa M. Mohammed. 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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