The purpose of this chapter is to introduce and study some new ideal convergence sequence spaces FSJθT, FS0JθT and FS∞JθT 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.
- Fuzzy number
- Lacunary sequence
- Compact operator
The concepts of fuzzy sets were initiated by Zadeh , since then it has become an active area of researchers. Matloka  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  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  studied fuzzy Ideals and fuzzy filters of pseudocomplemented semilattices Moreover, Nuray and Savas  extended the notion of convergence of the sequence of fuzzy real numbers to the notion of statistical convergence.
Fast  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  examined almost -statistical convergence of fuzzy numbers. Kostyrko et al.  introduced ideal convergence which is based on the natural density of the subsets of positive integers. Kumar and Kumar  extended the theory of ideal convergence to apply to sequences of fuzzy numbers. Khan et al. [10, 11, 12] studied the notion of -convergence in intuitionistic fuzzy normed spaces. Subsequently, Hazarika  studied the concept of lacunary ideal convergent sequence of fuzzy real numbers. Where a lacunary sequence is an increasing integer sequence such that and as . The intervals are determined by and defined by .
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.
In this section, we recall some basic notion, definitions and lemma that are required for the following sections.
It can be easily seen that is a partial order on and is a complete metric space. The absolute value of is defined by
Suppose be determined as
Hence, defines a metric on . The multiplicative and additive identity in are denoted by and , respectively.
for every one obtain ,
for every and every one obtain .
An ideal is known as non– trivial if and non– trivial ideal is said to be an admissible if .
for every one have ,
for every and one have .
The canonical pre-image of a sequence is a sequence defined as follows:
is in canonical pre-image of if is canonical pre-image of some element
That is, if for all infinite and the sequence , where for and otherwise, belongs to .
maps every bounded sequence in onto a sequence in which has a convergent subsequence.
3. Main results
In this section, we introduce the spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and investigate some topological and algebraic properties on these spaces. We denote the class of all sequences of fuzzy real numbers and be an admissible ideal of the subset of the natural numbers .
be such that . Therefore, the set
Thus, the sets on right hand side of (2) belong to . Therefore belongs to . Therefore, . Hence is linear space.
In similar manner, one can easily prove that and are linear spaces. □
Consider the sequence as:
For . Therefore
Hence, . Thus . Let be sequence in such a way that, for . Therefore, one obtain
For . Therefore
It can be easily seen that .
Hence and are not convergence free.
Fix an . Then, we have
Which holds for all . Hence
On Contrary, assume that
Then, the set
Let . If we fix an , then we have as well as . Hence . This implies that . That is
Where the diam of denotes the length of interval . Continuing in this way, by induction, we get the sequence of closed intervals.
With the property that
Then there exists a where in such a way that . Therefore, the result holds. □
That is, the set
On the both sides, taking the supremum over of the above equation, we obtain . Therefore inclusion holds.
Then, it proves that . Hence . □
Then . Applying Lemma 1, there exists a subset of in such a way that and . Determine a sequence as
Therefore belong to the canonical pre- image of the - step spaces of . But . Hence is not monotone. Hence, by Lemma (2.1) is not normal.
Suppose that sequence of scalars with the property . Therefore
Hence, from the Eq. (5) and above inequality, one obtain
Therefore, . Then is solid and monotone by lemma 2.1. □
Since be a Cauchy sequence in . Then for each there exists , such that
For a given ,
Then . Let . Where
Assume . Then, for every it has
Hence, is a Cauchy sequence of scalars in , so there exists scalar in such a way that as . For this step, let be given. Therefore it proved that whenever
Since , there exists so that
implies that . The number, can be chosen together with Eq. (7), it have
Which implies that . Since
Then it has a subset such that , where
Suppose that . Then, for every it has
In the following example to prove that is closed subspace of .
Hence, . Therefore and . So, it can be easily seen that is closed subspace of .
The spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and investigate algebraic and topological properties together with some examples on the given spaces. We proved that the new introduced sequence spaces are linear. Some spaces are convergent free and we also proved that space is closed subspace of . These new spaces and results provide new tools to help the authors for further research and to solve the engineering problems.