Intuitionistic anti fuzzy normed ideals

In this paper, we introduce the notion of intuitionistic anti fuzzy normed subrings and intuitionistic anti fuzzy normed ideals and various properties of intuitionistic anti fuzzy normal subrings and ideals are examined. Also, we prove that the intersection of two intuitionistic anti fuzzy normed ideals is also an intuitionistic anti fuzzy normed ideal. Besides, we define the anti level subsets and identify the relationship between these sets and intuitionistic anti fuzzy normed ideals. In addition, we define the anti-image of intuitionistic anti fuzzy normed ideals and discuss the inverse image and anti-image for intuitionistic anti fuzzy normed ideals under epimorphism mapping.

Intuitionistic fuzzy normed prime and maximal ideals

<abstract><p>Motivated by the new notion of intuitionistic fuzzy normed ideal, we present and investigate some associated properties of intuitionistic fuzzy normed ideals. We describe the intrinsic product of any two intuitionistic fuzzy normed subsets and show that the intrinsic product of intuitionistic fuzzy normed ideals is a subset of the intersection of these ideals. We specify the notions of intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal, we present the conditions under which a given intuitionistic fuzzy normed ideal is considered to be an intuitionistic fuzzy normed prime (maximal) ideal. In addition, the relation between the intuitionistic characteristic function and prime and maximal ideals is generalized. Finally, we characterize relevant properties of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals.</p></abstract>

Spectrum perturbations of compact operators in a Banach space

For an integer p ≥ 1, let Γp be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm NΓp(.) and the property $$\begin{array}{} \displaystyle \sum_{k=1}^{\infty} |\lambda_k(A)|^p\le a_p N_{{\it\Gamma}_p}^p(A) \;\;(A\in {\it\Gamma}_p), \end{array}$$ where λk(A) (k = 1, 2, …) are the eigenvalues of A and ap is a constant independent of A. Let A, Ã ∈ Γp and $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A):= N_{{\it\Gamma}_p}(A-\tilde A) \;\exp\;\left[a_p b_p^p \;\left(1+\frac{1}2 (N_{{\it\Gamma}_p}(A+\tilde A) + N_{{\it\Gamma}_p}(A-\tilde A))\right)^p\right], \end{array}$$ where bp is the quasi-triangle constant in Γp. It is proved the following result: let I be the unit operator, I – Ap be boundedly invertible and $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A)\exp\;\left[\frac{a_pN^p_{{\it\Gamma}_p}(A) } {\psi_p(A)}\right] \lt 1, \end{array}$$ where ψp(A) = infk=1,2,… |1 – $\begin{array}{} \displaystyle \lambda_k^{p} \end{array}$(A)|. Then I – Ãp is also boundedly invertible. Applications of that result to the spectrum perturbations of absolutely p-summing and absolutely (p, 2) summing operators are also discussed. As examples we consider the Hille-Tamarkin integral operators and matrices.

Fuzzy Normed Rings

In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal, and fuzzy normed maximal ideal of a normed ring, respectively. We show some algebraic properties of normed ring theory on fuzzy sets, prove theorems, and give relevant examples.