Research Article  Open Access
Nashat Faried, Mohamed S. S. Ali, Hanan H. Sakr, "On Fuzzy Soft Linear Operators in Fuzzy Soft Hilbert Spaces", Abstract and Applied Analysis, vol. 2020, Article ID 5804957, 13 pages, 2020. https://doi.org/10.1155/2020/5804957
On Fuzzy Soft Linear Operators in Fuzzy Soft Hilbert Spaces
Abstract
Due to the difficulty of representing problem parameters fuzziness using the soft set theory, the fuzzy soft set is regarded to be more general and flexible than using the soft set. In this paper, we define the fuzzy soft linear operator in the fuzzy soft Hilbert space based on the definition of the fuzzy soft inner product space in terms of the fuzzy soft vector modified in our work. Moreover, it is shown that , and are suitable examples of fuzzy soft Hilbert spaces and also some related examples, properties and results of fuzzy soft linear operators are introduced with proofs. In addition, we present the definition of the fuzzy soft orthogonal family and the fuzzy soft orthonormal family and introduce examples satisfying them. Furthermore, the fuzzy soft resolvent set, the fuzzy soft spectral radius, the fuzzy soft spectrum with its different types of fuzzy soft linear operators and the relations between those types are introduced. Moreover, the fuzzy soft right shift operator and the fuzzy soft left shift operator are defined with an example of each type on . In addition, it is proved, on , that the fuzzy soft point spectrum of fuzzy soft right shift operator has no fuzzy soft eigenvalues, the fuzzy soft residual spectrum of fuzzy soft right shift operator is equal to the fuzzy soft comparison spectrum of it and the fuzzy soft point spectrum of fuzzy soft left shift operator is the fuzzy soft open disk . Finally, it is shown that the fuzzy soft Hilbert space is fuzzy soft selfdual in this generalized setting.
1. Introduction
In real world, the complexity generally arises from uncertainty in the form of ambiguity. So, we always have many complicated problems in the areas like economics, engineering, medical science, environmental science, sociology, business management and many other fields. We cannot successfully use classical mathematical methods to overcome difficulties of uncertainties in those problems.
In 1965, Zadeh [1] proposed an extension of the set theory which is the theory of fuzzy sets to deal with uncertainty. Just as a crisp set on a universal set is defined by its characteristic function from to , a fuzzy set on a domain is defined by its membership (characteristic) function from to .
In 1999, Molodtsov [2] introduced an extension of the set theory namely soft set theory to overcome uncertainties and solve complicated problems which cannot be dealt with by classical methods in many areas such as Riemann integration, measure theory, environmental science, decisionmaking, game theory, physics, engineering, computer science, medicine, economics and many other fields. The soft set is a mathematical tool for modeling uncertainty by associating a set with a set of parameters, i.e., it is a parameterized family of subsets of the universal set. After that, many researchers introduced new extended concepts based on soft sets, gave examples for them and studied their properties like soft point [3], soft metric spaces [4], soft normed spaces [5], soft inner product spaces [6] and soft Hilbert spaces [7], etc. In addition, many researchers presented some applications of soft set theory in different areas (see [8–12]).
But almost all the time, although this progress, in real life problems and situations, we still have inexact information about our considered objects. So, to improve those two concepts; fuzzy set and soft set, Maji et al. [13] combined them together in one concept and called this new concept fuzzy soft set. This new concept widened the soft sets approach from crisp (ordinary) cases to fuzzy cases which is more general than any other. In recent years, many researchers applied this notion and gave some concepts such as fuzzy soft point [14], fuzzy soft real number [15], fuzzy soft metric spaces [16] and fuzzy soft normed spaces [17]. In addition, many researchers presented some applications of fuzzy soft set theory in different areas (see [18–25]).
In this work, we progress on these stated previous studies by introducing the fuzzy soft inner product on fuzzy soft vector spaces, the fuzzy soft CauchySchwartz inequality, the fuzzy soft orthogonality and the fuzzy soft Hilbert spaces. Moreover, we define the fuzzy soft linear operators in fuzzy soft Hilbert spaces, establish their related theorems, introduce fuzzy soft spectral theory of them and prove the fuzzy soft selfduality of fuzzy soft Hilbert spaces. The rest of the paper is organized as the following. Section 2 introduces the basic needed concepts and definitions. Section 3 studies the fuzzy soft linear operators in fuzzy soft Hilbert spaces and some related examples and properties. Furthermore, in section 3, fuzzy soft right shift operator, fuzzy soft left shift operator, fuzzy soft resolvent set, fuzzy soft spectral radius, fuzzy soft spectrum with its different types of them are investigated. Finally, in section 3, we show that the fuzzy soft Hilbert space is fuzzy soft selfdual. Section 4 provides conclusions and open questions for further investigations.
2. Definitions and Preliminaries
The aim of this section is to list some notations, definitions and preliminaries for fuzzy set, soft set and fuzzy soft set needed in the following discussion. In addition, it presents the fuzzy soft point definition including our present modification.
Definition 1 (Fuzzy set) [1]. Let be a universal set (space of points or objects). A fuzzy set (class) over is a set characterized by a function . is called the membership, characteristic or indicator function of the fuzzy set and the value is called the grade of membership of in .
Definition 2 (Soft set) ([2, 26]). A pair is said to be a soft set over a nonempty set provided that is a mapping of a set of parameters into . A soft set is identified as a set of ordered pairs: .
Example 3. Consider a soft set which describes “attractiveness of markets” under the consideration of a decision maker to purchase. Suppose that there are five markets to be considered in the universal set , denoted by and , where stands for the parameters in a word of “luxurious”, “expensive”, and “in a suitable location”, respectively. Thus, to define a soft set means to represent luxurious markets, expensive markets and so on. We can write the soft set over by the relation . This soft set can be represented in Table 1.
Definition 4 (Fuzzy soft set) [13]. Let be a universal set, be a set of parameters and . A pair is called a fuzzy soft set over , where is a mapping given by , is the family of all fuzzy subsets of (the power set of fuzzy sets on ) and the fuzzy subset of is defined as a map from to . The family of all fuzzy soft sets over a universal set , in which all the parameter sets are the same, is denoted by .
Example 5. Example (3) can be characterized by a membership function instead of numbers which associates each element with a real number in the interval , then we can write the fuzzy soft set over as . This fuzzy soft set can be represented in Table 2.


The following definition and its consequent related definitions take their present formula according to our modification as follows:
Definition 6 (Fuzzy soft point) [14]. The fuzzy soft set is called a fuzzy soft point over , denoted by (briefly denoted by ), if for the element and ( is the value of the membership degree), The fuzzy soft point can be considered as the quadruple .
Example 7. Suppose that there are four points in the universal set , i.e., and let be the set of parameters. Then, is a fuzzy soft point over .
Definition 8 (The complement of a fuzzy soft point) [27]. The fuzzy soft point is called the fuzzy soft complement of a fuzzy soft point , denoted by , if for the element and ,
Example 9. The complement of the fuzzy soft point stated in Example (7) is the fuzzy soft point .
The collection of all fuzzy soft points over is denoted by . denotes the set of all fuzzy soft real numbers and denotes the set of all nonnegative fuzzy soft real numbers (such as , in symbol). denotes the set of all fuzzy soft complex numbers (such as , in symbol). Note that the fuzzy soft zero vector and the fuzzy soft unit vector .
Definition 10 (Fuzzy soft vector space)([17]). Let be a vector space over a field and the parameter set be the set of all real numbers and . The fuzzy soft set is called a fuzzy soft vector over , denoted by (briefly denoted by ), if there is exactly one such that for some and for all ( is the value of the membership degree). The set of all fuzzy soft vectors over is denoted by . The set is said to be a fuzzy soft vector space or a fuzzy soft linear space of over if is a vector subspace of , for all . is a fuzzy soft vector space according to the following two operations: (1), for all (2), for all and for all .
Example 11. Consider the Euclidean dimensional space over .
Let be the set of parameters. Let be defined as follows:
; coordinate of is }, .
Then is a fuzzy soft vector space or a fuzzy soft linear space of over .
In addition, Let be a fuzzy soft element of as follows:
, . Then is a fuzzy soft vector of .
Definition 12 (Fuzzy soft metric space) [16]. A fuzzy soft metric space is a fuzzy soft set with a fuzzy soft realvalued function satisfying the fuzzy soft metric conditions as the following:
(FSM1) , for all , and .
(FSM2) , for all .
(FSM3) , for all .
Definition 13 (Fuzzy soft normed space) [17]. Let be a fuzzy soft vector space. Then, a mapping is said to be a fuzzy soft norm on if satisfies the following conditions:
(FSN1) , for all , and .
(FSN2) , for all and for all fuzzy soft scalar .
(FSN3) , for all .
3. Main Results
The aim of this section is to introduce the fuzzy soft inner product spaces, the fuzzy soft Hilbert spaces and the fuzzy soft linear operators in fuzzy soft Hilbert spaces and to study some theorems and results of them. In addition, fuzzy soft resolvent set, fuzzy soft spectral radius, fuzzy soft spectrum with its different types of them and more results are established. Furthermore, fuzzy soft right shift operator and fuzzy soft left shift operator are defined. Finally, the fuzzy soft selfduality of fuzzy soft Hilbert space is introduced.
Definition 14 (Fuzzy soft inner product space) [28]. Let be a fuzzy soft vector space. Then, the mapping is said to be a complex fuzzy soft inner product (shortly, fuzzy soft inner product) on if satisfies the following axioms:
(FSI1) , for all , and .
(FSI2) , for all , where bar denotes the complex conjugate of the fuzzy soft complex number.
(FSI3) , for all and for all fuzzy soft scalar .
(FSI4) , for all .
The fuzzy soft vector space with a fuzzy soft inner product is said to be a complex fuzzy soft inner product space (shortly, fuzzy soft inner product space) and is denoted by .
Definition 15 (Real fuzzy soft inner product space) [28]. If the mapping in the above Definition (14) is replaced by , then it is called a real fuzzy soft inner product space and its Conditions (Axioms (FSI1),(FSI3),(FSI4)) are the same, but the Condition (FSI2) is replaced by
(FSI2(i)): , for all .
Theorem 16 (Fuzzy soft CauchySchwartz inequality) [28]. Let be a fuzzy soft inner product space, then for all , we have
Theorem 17 (see [28]). For all , a fuzzy soft inner product space can be considered as a fuzzy soft normed space with .
Example 18. the fuzzy soft complex Euclidean space (the space of all fuzzy soft dimensional complex numbers) is a complex fuzzy soft inner product space with the complex fuzzy soft inner product defined as follows: for all .
Solution. The proof is straightforward by applying the conditions of the complex fuzzy soft inner product space stated in Definition (14).
Example 19. the fuzzy soft real Euclidean space (the space of all fuzzy soft dimensional real numbers) is a real fuzzy soft inner product space with the real fuzzy soft inner product defined as follows: for all .
Solution. The proof is straightforward by applying the conditions of the real fuzzy soft inner product space stated in Definition (15).
Example 20. the space of all fuzzy soft squaresummable sequences is a complex fuzzy soft inner product space with the complex fuzzy soft inner product defined as follows: for all .
Solution. The proof is straightforward by applying the conditions of the complex fuzzy soft inner product space stated in Definition (14).
Example 21. the space of all fuzzy soft complexvalued continuous functions on is a complex fuzzy soft inner product space with the complex fuzzy soft inner product defined as follows: for all .
Solution. The proof is straightforward by applying the conditions of the complex fuzzy soft inner product space stated in Definition (14).
Theorem 22 (Fuzzy soft polarization identity) [28]. Let be a complex fuzzy soft inner product space and let . Then, we can write the fuzzy soft polarization identity in the following formula: for all .
Definition 23 (Fuzzy soft Hilbert space) [29]. Let be a fuzzy soft inner product space. Then, this space, which is fuzzy soft complete in the induced fuzzy soft norm stated in Theorem (17), is called a fuzzy soft Hilbert space, denoted by (shortly ). It is clear that every fuzzy soft Hilbert space is a fuzzy soft Banach space.
Example 24. The space the fuzzy soft complex Euclidean space (the space of all fuzzy soft dimensional complex numbers) is a complex fuzzy soft Hilbert space with the complex fuzzy soft inner product defined as follows: for all .
Solution. We have is an fuzzy soft inner product space from Example (18). Using Theorem (17), we obtain: and since is fuzzy soft complete in this fuzzy soft norm. Then, is an fuzzy soft complete fuzzy soft inner product space, i.e., an fuzzy soft Hilbert space.
Example 25. The space the fuzzy soft real Euclidean space (the space of all fuzzy soft dimensional real numbers) is an fuzzy soft Hilbert space with the real fuzzy soft inner product defined as follows: for all .
Solution. The proof is easy by using Example (19) and Theorem (17), similarly as Example (24).
Example 26. The space the space of all fuzzy soft squaresummable sequences is an fuzzy soft Hilbert space with the complex fuzzy soft inner product defined as follows: for all .
Solution. The space is fuzzy soft complete as an fuzzy soft normed space. But is an fuzzy soft inner product space from Example (20). Then, is an fuzzy soft Hilbert space.
Definition 27 (Fuzzy soft orthogonal family) [29]. Let be a fuzzy soft inner product space. A family of fuzzy soft elements of is called a fuzzy soft orthogonal family if
Definition 28 (Fuzzy soft orthonormal family) [29]. Let be a fuzzy soft inner product space. A family of fuzzy soft elements of is called a fuzzy soft orthonormal family if is a fuzzy soft orthogonal family (i.e., satisfies the Condition (13)) and .
Remark 29 (see [29]). Note that if is a fuzzy soft orthonormal family, then , where
Example 30. The fuzzy soft set of fuzzy soft elements , and is a fuzzy soft orthonormal family in .
Solution. Let , and . It is clear that ; .
Example 31. The fuzzy soft set of fuzzy soft elements , and , is a fuzzy soft orthonormal family in .
Solution. Let , and ,. It is clear that ; .
Example 32. Let be the space of all fuzzy soft complexvalued continuous functions on . For , let be defined by: Then, is a fuzzy soft orthonormal family in .
Solution. Hence, , where is defined as in (15), is a fuzzy soft orthonormal family in .
Definition 33 (Fuzzy soft linear operator). Let be an operator. Then is said to be fuzzy soft linear if
(1), For all fuzzy soft elements (additivity).(2), for all fuzzy soft element and for all fuzzy soft scalar (homogeneity).The Conditions (1, 2) can be combined in one condition as follows:
, for all fuzzy soft elements and for all fuzzy soft scalars .
Definition 34 (Fuzzy soft linear operator in ). Let be a fuzzy soft Hilbert space. A fuzzy soft linear operator is called a fuzzy soft linear operator in . If, then is a fuzzy soft linear operator on , written . If is fuzzy soft bounded (i.e., : ), then .
Example 35. The fuzzy soft identity operator defined by , for all , is a fuzzy soft linear operator on .
Example 36. The fuzzy soft zero operator defined by , for all , is a fuzzy soft linear operator on .
Definition 37 (Fuzzy soft right shift operator). Let be a fuzzy soft Hilbert space. If . Define the fuzzy soft operator as follows: The fuzzy soft operator is called the fuzzy soft right shift operator.
Example 38. The fuzzy soft right shift operator defined as above in Definition (37) is a fuzzy soft linear operator on .
Definition 39 (Fuzzy soft left shift operator). Let be a fuzzy soft Hilbert space. If . Define the fuzzy soft operator as follows: The fuzzy soft operator is called the fuzzy soft left shift operator.
Example 40. The fuzzy soft left shift operator defined as above in Definition 3.9 is a fuzzy soft linear operator on .
Definition 41 (Fuzzy soft adjoint operator in ). The fuzzy soft adjoint operator of a fuzzy soft linear operator is defined by , for all .
Example 42. The fuzzy soft left shift operator defined in Definition (39) is the fuzzy soft adjoint of the fuzzy soft right shift operator defined in Definition (37).
Example 43. Let be the space of all fuzzy soft squaresummable sequences. If . By applying the fuzzy soft operator (17) from the above Definition (37), we can define the fuzzy soft right shift operator (the fuzzy soft unilateral shift operator) in as follows: If is a fuzzy soft orthonormal basis for , i.e., the set of fuzzy soft elements is a fuzzy soft orthonormal family (fuzzy soft orthonormal basis) in . Then, by using (19), we get: For the fuzzy soft adjoint , we have: Therefore, by using Formula (21) in (22), the following is obtained: Since then If , then we have: Thus, by substituting from (26) in (23), we get: and therefore, the following formula is obtained: Now, by applying (25) and (28), we have: Then, by substituting from (20) in (29), we obtain: Hence, we finally get: Equation (31) represents the fuzzy soft adjoint of fuzzy soft unilateral shift operator, which is usually said to be a fuzzy soft left shift operator.
Theorem 44. Let be a fuzzy soft inner product space and . Then, we have , where is the fuzzy soft range of and is the fuzzy soft null (fuzzy soft kernel) of