**Abstract**

The purpose of this chapter is to introduce and study some new ideal convergence sequence spaces *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* ð Þ T , *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>0</sup> ð Þ T and *<sup>F</sup>*S<sup>J</sup> *<sup>θ</sup>* <sup>∞</sup> ð Þ T on a fuzzy real number *F* defined by a compact operator T . We investigate algebraic properties like linearity, solidness and monotinicity with some important examples. Further, we also analyze closedness of the subspace and inclusion relations on the said spaces.

**Keywords:** Ideal, J{convergence, J{Cauchy, Fuzzy number, Lacunary sequence, Compact operator

## **1. Introduction**

The concepts of fuzzy sets were initiated by Zadeh [1], since then it has become an active area of researchers. Matloka [2] initiated the notion of ordinary convergence of a sequence of fuzzy real numbers and studied convergent and bounded sequences of fuzzy numbers and some of their properties, and proved that every convergent sequence of fuzzy numbers is bounded. Nanda [3] investigated some basic properties for these sequences and showed that the set of all convergent sequences of fuzzy real numbers form a complete metric space. Alaba and Norahun [4] studied fuzzy Ideals and fuzzy filters of pseudocomplemented semilattices Moreover, Nuray and Savas [5] extended the notion of convergence of the sequence of fuzzy real numbers to the notion of statistical convergence.

Fast [6] introduced the theory of statistical convergence. After that, and under different names, statistical convergence has been discussed in the ergodic theory, Fourier analysis and number theory. Furthermore, it was examined from the sequence space point of view and linked with summability theory. Esi and Acikgoz [7] examined almost *λ*-statistical convergence of fuzzy numbers. Kostyrko et al. [8] introduced ideal J �convergence which is based on the natural density of the subsets of positive integers. Kumar and Kumar [9] extended the theory of ideal convergence to apply to sequences of fuzzy numbers. Khan et al. [10–12] studied the notion of J -convergence in intuitionistic fuzzy normed spaces. Subsequently, Hazarika [3] studied the concept of lacunary ideal convergent sequence of fuzzy real numbers. Where a lacunary sequence is an increasing integer sequence *θ* ¼ ð Þ *kr* such that *k*<sup>0</sup> ¼ 0 and *hr* ¼ *kr* � *kr*�<sup>1</sup> ! ∞ as *r* ! ∞. The intervals are determined by *θ* and defined by *Ir* ¼ ð � *kr*�1, *kr* .

We outline the present work as follows. In Section 2, we recall some basic definitions related to the fuzzy number, ideal convergent, monotonic sequence and compact operator. In Section 3, we introduce the spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and prove our maim results. In Section 4, we state the conclusion of this chapter.
