Journal of Applied Science and Engineering

Published by Tamkang University Press


Impact Factor



Laith K. Shaakir1, Manaf Adnan Saleh Saleh This email address is being protected from spambots. You need JavaScript enabled to view it.2, and Ranen Z. Ahmood2

1Department of Mathematics, College of Computer Sciences and Mathematics, Tikrit University, Tikrit, Iraq
2Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, Baghdad, Iraq


Received: February 11, 2022
Accepted: April 19, 2022
Publication Date: June 10, 2022

 Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Download Citation: ||  


Although the systematic emergence of the fuzzy functional analysis theory has started in the last few years, we are starting to construct a new theory of fuzzy operator ideals inspired by the classical (crisp) operator ideals theory. We define the concept of fuzzy operator ideal and present some basic examples and introduce a significant class of fuzzy operator ideals, namely the class of absolutely fuzzy p-summing between arbitrary complete fuzzy normed spaces, which is a natural generalization of the notion of absolutely (crisp) p-summing between Banach spaces defined by Albrecht Pietsch. We initiate to define fuzzy norm of the aforementioned notion and prove its fuzzy norm is fuzzy real number. It shows that the resulting class of fuzzy bounded operators is a complete fuzzy normed fuzzy operator ideal. It then establishes an important fundamental characterization of absolutely fuzzy p-summing. This is done by proving a fuzzy version of Pietsch Domination Theorems. Finally, the paper presents some open problems which the researchers finds interesting.

Keywords: Operator ideals, Fuzzy functional analysis, Fuzzy real analysis, Pietsch Domination Theorem


  1. [1] J. Xiao and X. Zhu, (2002) “On linearly topological structure and property of fuzzy normed linear space" Fuzzy sets and Systems 125(2): 153–161. DOI: 10.1016/S0165-0114(00)00136-6.
  2. [2] C. Felbin, (1992) “Finite dimensional fuzzy normed linear space" Fuzzy sets and Systems 48(2): 239–248. DOI: 10.1016/0165-0114(92)90338-5.
  3. [3] O. Kaleva and S. Seikkala, (1984) “On fuzzy metric spaces" Fuzzy sets and Systems 12(3): 215–229. DOI: 10.1016/0165-0114(84)90069-1.
  4. [4] A. N. A. Hasankhani and M. Saheli, (2010) “Some properties of fuzzy Hilbert spaces and norm of operators" Iranian Journal of Fuzzy Systems 7(3): 129–157. DOI: 10.22111/IJFS.2010.196.
  5. [5] B. Daraby and J. Jafari, (2016) “Some properties of fuzzy real numbers" Sahand Communications in Mathematical Analysis 3(1): 21–27.
  6. [6] Z. Feiyue, (1992) “On fuzzy-valued inequalities" Fuzzy sets and Systems 45(2): 215–218. DOI: 10.1016/0165-0114(92)90120-S.
  7. [7] A. Pietsch. Operator ideals. North-Holland Publishing Company, 1980.
  8. [8] E. D. D. Achour and M. A. S. Saleh, (2018) “Multilinear mixing operators and Lipschitz mixing operator ideals" Oper. Matrices 12(4): 903–931. DOI: 10.7153/oam-2018-12-55.
  9. [9] M. A. S. Saleh, (2017) “New types of Lipschitz summing maps between metric spaces" Mathematische Nachrichten 290(8-9): 1347–1373. DOI: 10.1002/mana.201500020.
  10. [10] M. A. S. Saleh, (2021) “Interpolative Lipschitz ideals" Colloquium Mathematicum 163(1): 153–170. DOI:10.4064/cm7844-10-2019.
  11. [11] H. Jarchow. Locally convex spaces. Springer Science and Business Media, 2012.
  12. [12] J. Farmer and W. Johnson, (2009) “Lipschitz psumming operators" Proceedings of the American Mathematical Society 137(9): 2989–2995. DOI: 10.1090/S0002-9939-09-09865-7.
  13. [13] M. A. S. Saleh, (2021) “Computations of Lipschitz summing norms and applications" Colloquium Mathematicum 165(1): 31–40. DOI: 10.4064/cm8096-3-2020.