On Fuzzy Soft Linear Operators in Fuzzy Soft Hilbert Spaces

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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 X is defined by its characteristic function from X to f0, 1g, a fuzzy set on a domain X is defined by its membership (characteristic) function from X to ½0, 1.
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 integra-tion, measure theory, environmental science, decision-making, 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][9][10][11][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][19][20][21][22][23][24][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 Cauchy-Schwartz 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 self-duality 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 self-dual.Section 4 provides conclusions and open questions for further investigations.

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 U be a universal set (space of points or objects).A fuzzy set (class) X over U is a set characterized by a function f X : U → ½0, 1. f X is called the membership, characteristic or indicator function of the fuzzy set X and the value f X ðuÞ is called the grade of membership of u ∈ U in X. Definition 2 (Soft set) ( [2,26]).A pair ðU, EÞ is said to be a soft set over a non-empty set X provided that U is a mapping of a set of parameters E into 2 U .A soft set is identified as a set of ordered pairs: ðU, EÞ = fðe, UðeÞÞ: e ∈ E and GðeÞ ∈ 2 U g. Example 3. Consider a soft set ðF, AÞ 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 U, denoted by U = fm 1 , m 2 , m 3 , m 4 , m 5 g and A = fa 1 , a 2 , a 3 g, where a i ði = 1, 2, 3Þ 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 ðF, AÞ over U by the relation ðF, AÞ = fða 1 , fm 2 , m 4 gÞ, ða 2 , fm 5 gÞ, ða 3 , fm 1 , m 3 , m 4 gÞg.This soft set can be represented in Table 1.
Definition 4 (Fuzzy soft set) [13].Let U be a universal set, E be a set of parameters and A ⊆ E. A pair ðG, AÞ is called a fuzzy soft set over U, where G is a mapping given by G : A ⟶ FðUÞ, FðUÞ is the family of all fuzzy subsets of U (the power set of fuzzy sets on U) and the fuzzy subset of U is defined as a map f from U to ½0, 1.The family of all fuzzy soft sets ðG, AÞ over a universal set U, in which all the parameter sets A are the same, is denoted by FSS ðUÞ A = FSSð ŨÞ.
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 ðG, AÞ ∈ FSSð ŨÞ is called a fuzzy soft point over U, denoted by ðu f GðeÞ , AÞ (briefly denoted by ũf GðeÞ ), if for the element e ∈ A and u ∈ U (α ∈ ð0, 1 is the value of the membership degree), The fuzzy soft point can be considered as the quadruple ðu 0 , e 0 , G, αÞ.
Example 7. Suppose that there are four points in the universal set U , i.e., U = fu 1 , u 2 , u 3 , u 4 g and let A = fe 1 , e 2 , e 3 , e 4 , e 5 g Example 9.The complement of the fuzzy soft point stated in Example (7) is the fuzzy soft point ðu 2 , e 5 , G, 0:4Þ = ũ2 0:4 Gðe 5 Þ .The collection of all fuzzy soft points over U is denoted by FSPðUÞ A = FSPð ŨÞ.ℝðAÞ denotes the set of all fuzzy soft real numbers and ℝ + ðAÞ denotes the set of all non-negative fuzzy soft real numbers (such as r, in symbol).ℂðAÞ denotes the set of all fuzzy soft complex numbers (such as c, in symbol).Note that the fuzzy soft zero vector θ = ð 0, 0, 0, 0Þ and the fuzzy soft unit vector j = ð 1, 1, 1, 1Þ.
Definition 10 (Fuzzy soft vector space)( [17]).Let U be a vector space over a field KðK = ℝÞ and the parameter set E be the set of all real numbers ℝ and A ⊆ E. The fuzzy soft set ðG, AÞ ∈ FSSð ŨÞ is called a fuzzy soft vector over U, denoted by ðv f GðeÞ , AÞ (briefly denoted by ṽf GðeÞ ), if there is exactly one e ∈ A such that f GðeÞ ðvÞ = α for some v ∈ U and f Gðe ′ ÞðvÞ = 0 for all e ′ ∈ A − feg ( α ∈ ð0, 1 is the value of the membership degree).The set of all fuzzy soft vectors over U is denoted by FSVðUÞ A = FSVð ŨÞ.The set Ṽ = FSVð ŨÞ is said to be a fuzzy soft vector space or a fuzzy soft linear space of U over K if ṼðeÞ is a vector subspace of U, for all e ∈ A. Ṽ is a fuzzy soft vector space according to the following two operations: (1) ṽ1 (2) r • ṽf GðeÞ = ð rvÞ f GðreÞ , for all ṽf GðeÞ ∈ Ṽ and for all r∈ℝðAÞ.
Example 11.Consider the Euclidean n − dimensional space ℝ n over ℝ.
Let A = f1, 2, 3,⋯,ng be the set of parameters.Let Ṽ : A ⟶ Fðℝ n Þ be defined as follows: ∈ℝ n ðAÞ; i th coordinate of ṽn Then Ṽ is a fuzzy soft vector space or a fuzzy soft linear space of ℝ n over ℝ.
In addition, Let ṽn be a fuzzy soft element of Ṽ as follows: Then ṽn is a fuzzy soft vector of Ṽ.

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.
3 Abstract and Applied Analysis (FSI2) ∈FSVð ŨÞ, where bar denotes the complex conjugate of the fuzzy soft complex number.
∈FSVð ŨÞ and for all fuzzy soft scalar c.
The fuzzy soft vector space FSVð ŨÞ with a fuzzy soft inner product g <•, • > is said to be a complex fuzzy soft inner product space (shortly, fuzzy soft inner product space) and is denoted by ð Ũ, g h•, • iÞ.
Example 18. ℂ n ðAÞ the fuzzy soft complex Euclidean space (the space of all fuzzy soft n − dimensional complex numbers) is a complex fuzzy soft inner product space with the complex fuzzy soft inner product defined as follows: for all ṽf GðeÞ , ũg GðaÞ ∈ℂ n ðAÞ.
Solution.The proof is straightforward by applying the conditions of the complex fuzzy soft inner product space stated in Definition ( 14).
Example 19.ℝ n ðAÞ the fuzzy soft real Euclidean space (the space of all fuzzy soft n − dimensional real numbers) is a real fuzzy soft inner product space with the real fuzzy soft inner product defined as follows: for all ṽf GðeÞ , ũg GðaÞ ∈ℝ n ðAÞ.
Solution.The proof is straightforward by applying the conditions of the real fuzzy soft inner product space stated in Definition (15).
Example 20. ℓ 2 ðAÞ 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 ṽf GðeÞ , ũg GðaÞ ∈ℓ 2 ðAÞ.
Solution.The proof is straightforward by applying the conditions of the complex fuzzy soft inner product space stated in Definition (14).
Example 21.C ½ 0, 1 ðAÞ the space of all fuzzy soft complexvalued continuous functions on ½ 0, 1 is a complex fuzzy soft inner product space with the complex fuzzy soft inner product defined as follows: for all ηf GðeÞ , ξg GðaÞ ∈ C ½ 0, 1 ðAÞ.
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 ð Ũ, g h•, • iÞ be a complex fuzzy soft inner product space and let ṽ1 ∈FSVð ŨÞ.Then, we can write the fuzzy soft 4 Abstract and Applied Analysis polarization identity in the following formula: Definition 23 (Fuzzy soft Hilbert space) [29].Let ð Ũ, g h•, • iÞ 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 ð H, g h•, • iÞ (shortly H).It is clear that every fuzzy soft Hilbert space is a fuzzy soft Banach space.
Example 24.The space ℂ n ðAÞ the fuzzy soft complex Euclidean space (the space of all fuzzy soft n − dimensional complex numbers) is a complex fuzzy soft Hilbert space with the complex fuzzy soft inner product defined as follows: for all ṽf GðeÞ , ũg GðaÞ ∈ℂ n ðAÞ.
Solution.We have ℂ n ðAÞ is an fuzzy soft inner product space from Example (18).Using Theorem (17), we obtain: and since ℂ n ðAÞ is fuzzy soft complete in this fuzzy soft norm.Then, ℂ n ðAÞ is an fuzzy soft complete fuzzy soft inner product space, i.e., an fuzzy soft Hilbert space.
Example 25.The space ℝ n ðAÞ the fuzzy soft real Euclidean space (the space of all fuzzy soft n − dimensional real numbers) is an fuzzy soft Hilbert space with the real fuzzy soft inner product defined as follows: for all ṽf GðeÞ , ũg GðaÞ ∈ℝ n ðAÞ.
Solution.The proof is easy by using Example (19) and Theorem (17), similarly as Example (24).
Example 26.The space ℓ 2 ðAÞ the space of all fuzzy soft square-summable sequences is an fuzzy soft Hilbert space with the complex fuzzy soft inner product defined as follows: for all ṽf GðeÞ , ũg GðaÞ ∈ℓ 2 ðAÞ.
Solution.The space ℓ 2 ðAÞ is fuzzy soft complete as an fuzzy soft normed space.But ℓ 2 ðAÞ is an fuzzy soft inner product space from Example (20).Then, ℓ 2 ðAÞ is an fuzzy soft Hilbert space.
Definition 27 (Fuzzy soft orthogonal family) [29].Let H be a fuzzy soft inner product space.A family fṽ i elements of H is called a fuzzy soft orthogonal family if , ṽj Definition 28 (Fuzzy soft orthonormal family) [29].Let H be a fuzzy soft inner product space.A family fṽ i g is a fuzzy soft orthogonal family (i.e., satisfies the Condition (13)) and g kṽ i Remark 29 (see [29]).Note that if fṽ i Example 30.The fuzzy soft set of fuzzy soft elements ð j, θ, θÞ, ð θ, j, θÞ and ð θ, θ, jÞ is a fuzzy soft orthonormal family in ℝ 3 ðAÞ.
Hence, fẽ k g, where fẽ k g is defined as in (15), is a fuzzy soft orthonormal family in C ½ 0, 1 ðAÞ.
The Conditions (1, 2) can be combined in one condition as follows: Tðc 1 ṽ1 Þ, for all fuzzy soft elements ṽ1 Definition 37 (Fuzzy soft right shift operator).Let H be a fuzzy soft Hilbert space.If ṽf GðeÞ = ðṽ 1 , ṽ3 Define the fuzzy soft operator R as follows: , ṽ2 , ṽ3 , ṽ2 The fuzzy soft operator R is called the fuzzy soft right shift operator.
Example 38.The fuzzy soft right shift operator R : H ⟶ H defined as above in Definition (37) is a fuzzy soft linear operator on H.
Definition 39 (Fuzzy soft left shift operator).Let H be a fuzzy soft Hilbert space.If ṽf GðeÞ = ðṽ 1 the fuzzy soft operator L as follows: , ṽ3 The fuzzy soft operator L is called the fuzzy soft left shift operator.
Example 40.The fuzzy soft left shift operator L : H ⟶ H defined as above in Definition 3.9 is a fuzzy soft linear operator on H.
Definition 41 (Fuzzy soft adjoint operator in H).The fuzzy soft adjoint operator T * of a fuzzy soft linear operator T is defined by hT ṽ1 Example 42.The fuzzy soft left shift operator L defined in Definition (39) is the fuzzy soft adjoint of the fuzzy soft right shift operator R defined in Definition (37).
Theorem 46.Let T∈ Lð HÞ, where H is a fuzzy soft inner product space.If g k Tṽ f GðeÞ k = g kṽ f GðeÞ k for all ṽf GðeÞ ∈ H, then Proof.By using Theorem (22), we have: Now, using the given that g k Tṽ f GðeÞ k = g kṽ f GðeÞ k for all ṽf GðeÞ ∈ H, we obtain that: Theorem 47.Let T∈ Bð HÞ, where H is a fuzzy soft Hilbert space.Then, T * is fuzzy soft bounded and Proof.Using Theorem (17) and Theorem ( 16), we have: for all ṽf GðeÞ ∈ H. Then g k T * ṽ f GðeÞ k ≤ g k Tk g kṽ f GðeÞ k for all ṽf GðeÞ ∈ H. Therefore, and thus T * is fuzzy soft bounded.Similarly, using (1) from Theorem (45), we get: Hence, from (37), (38), we obtain that Theorem 48.Let T∈ Bð HÞ, where H is a fuzzy soft Hilbert space.Then, Proof.We have for all ṽf GðeÞ ∈ H: By using (35) from Theorem (47) in (39), we obtain that 8 Abstract and Applied Analysis g k T * Tṽ f GðeÞ k ≤ g k Tk 2 g kṽ f GðeÞ k for all ṽf GðeÞ ∈ H. Hence, Conversely, using Theorem (17), we get for all ṽf GðeÞ ∈ H: . That is to say that Therefore, from (40), (42), the following result is obtained: Now, by applying this result (43) for T * in place of T, we Definition 49 (Fuzzy soft resolvent set of T).Let T∈ Bð HÞ, where H is a fuzzy soft Hilbert space.Then, we define ρð TÞ = f λ∈ℂðAÞ: ð λĨ − TÞ −1 exists and fuzzy soft bounded} = f λ ∈ℂðAÞ: j λj> k Tkg.
We call ρð TÞ the fuzzy soft resolvent set of a fuzzy soft linear operator T.
Definition 50 (Fuzzy soft spectrum of T).Let T∈ Bð HÞ, where H is a fuzzy soft Hilbert space.Then, we define σð TÞ = ρ ð TÞ C = f λ∈ℂðAÞ: j λj ≤ g k Tkg.We call σð TÞ the fuzzy soft spectrum of a fuzzy soft linear operator T.
Definition 51 (Fuzzy soft spectral radius of T).Since, for T∈ Bð HÞ, where H is a fuzzy soft Hilbert space, σð TÞ is nonempty fuzzy soft compact (fuzzy soft closed and fuzzy soft bounded), we define rσ ð TÞ = sup fj λj: λ∈σð TÞg.We call rσ ð TÞ the fuzzy soft spectral radius of a fuzzy soft linear operator T. Also, we can show that rσ ð TÞ = lim Definition 52 (Fuzzy soft point spectrum of T). λ∈ℂðAÞ is said to be a fuzzy soft eigenvalue of a fuzzy soft linear operator T if λĨ − T is not fuzzy soft injective, i.e., there exists a non-zero fuzzy soft element ṽf GðeÞ ∈ H such that Tṽ f GðeÞ = λ ṽf GðeÞ .Moreover, ṽf GðeÞ ≠ θ is called the fuzzy soft eigenvector of a fuzzy soft linear operator T corresponding to λ.The set of all such λ is called the fuzzy soft point spectrum of a fuzzy soft linear operator T, denoted by σp ð TÞ.We have σp ð TÞ⊂σð TÞ.
Definition 53 (Fuzzy soft approximate point spectrum of T). λ∈ℂðAÞ is said to be a fuzzy soft approximate eigenvalue of a fuzzy soft linear operator T if there exists a fuzzy soft sequence of fuzzy soft elements fṽ n The set of all such λ is called the fuzzy soft approximate point spectrum of a fuzzy soft linear operator T, denoted by σa ð TÞ.We have σa ð TÞ⊂σð TÞ.
Definition 54 (Fuzzy soft comparison spectrum of T).The set of all λ∈ℂðAÞ such that the fuzzy soft range of λĨ − T is not fuzzy soft dense in H , (i.e. Rð λĨ TÞ ≠ H) is called the fuzzy soft comparison spectrum of a fuzzy soft linear operator T, denoted by σCom ð TÞ.
Definition 55 (Fuzzy soft residual spectrum of T).The set of all λ∈ℂðAÞ such that λĨ − T is fuzzy soft injective, but its fuzzy soft range is not fuzzy soft dense in H is called the fuzzy soft residual spectrum of a fuzzy soft linear operator T, denoted by σr ð TÞ.That is to say that: Definition 56 (Fuzzy soft continuous spectrum of T).The set of all λ∈ℂðAÞ such that λĨ − T is fuzzy soft injective and has fuzzy soft dense range in H, but is fuzzy soft singular (has no fuzzy soft inverse) is called the fuzzy soft continuous spectrum of a fuzzy soft linear operator T, denoted by σc ð TÞ.It is clear that σp ð TÞ ∪σ c ð TÞ⊂σ a ð TÞ, and the terms on the right being mutually disjoint.
Theorem 57.Let H be a fuzzy soft Hilbert space and T∈ Bð HÞ.
Proof.Let λ∈σ a ð TÞ, then there exists a fuzzy soft sequence of fuzzy soft elements fṽ n . Then, 9 Abstract and Applied Analysis (2) T and T * have the same fuzzy soft spectrum, i.e., σð TÞ = σð T * Þ.
That is to say that, Conversely, we want to prove that every fuzzy soft linear continuous functional f∈ H * defines a unique fuzzy soft element wh GðeÞ ∈ H with the same fuzzy soft norm (i.e., to prove that g k wh GðeÞ k ≤ f k fk).Let f∈ H * be a fuzzy soft linear continuous functional on H, and let B = ker f= fṽ f GðaÞ ∈ H : fðṽ f GðaÞ Þ = θg.
To prove that B is a fuzzy soft subspace of H, we have first to prove that B is a fuzzy soft linear manifold of H and second we have to prove that B is fuzzy soft closed.First, to prove that B is a fuzzy soft linear manifold of H, let ṽ1 ∈ B and let α, β be fuzzy soft scalars.Then, fðṽ 1 Therefore, we have αṽ 1 Hence, B is a fuzzy soft linear manifold of H. Second, to prove that B is fuzzy soft closed, let ṽn = ðz q GðdÞ /ηÞ.Therefore, If ṽf GðaÞ is any fuzzy soft point of H, then Thus, by substituting from (63) in (64), we obtain that fðṽ f GðaÞ Þ = ξf ðṽ 1 Hence, ṽf GðaÞ = ũg GðbÞ + ξṽ 1 Therefore, we have: Then, g hṽ f GðaÞ , ðṽ 1 ṽf GðaÞ ≠θ ðj fðṽ f GðaÞ Þj/ g kṽ f GðaÞ k Þ ≥ ðj fð wh GðeÞ Þj/ g k wh GðeÞ kÞ = ð g h wh GðeÞ , wh GðeÞ i/ g k wh GðeÞ kÞ = ð g k wh GðeÞ k g k wh GðeÞ k/ g k wh GðeÞ kÞ = g k wh GðeÞ k.That is to say that, Hence, from (61) and (66), we obtain that k fk = g k wh GðeÞ k.
At the end, we have to show that the fuzzy soft element wh GðeÞ is unique.Suppose that

Conclusions and Future Work
The fuzzy version or the soft version of topics like metric spaces, normed spaces, Banach spaces, operators, dual spaces, functionals, inner product spaces, Hilbert spaces, operators on Hilbert spaces and many other topics have been studied by many mathematicians.On the other hand, combining fuzzy and soft sets together is not only more general than using only one of them but also gives us more extended and accurate results.Few researchers have studied some of those general extensions concepts such as fuzzy soft normed spaces and fuzzy soft metric spaces.We have defined the fuzzy soft inner product space ð Ũ, g h•, • iÞ by using the concept of fuzzy soft vector ṽf GðeÞ modified in our work and have shown that ℂ n ðAÞ, ℝ n ðAÞ, ℓ 2 ðAÞ and C ½ 0, 1 ðAÞ are suitable examples of fuzzy soft inner product spaces.Also, the definition of the fuzzy soft Hilbert space H with ℂ n ðAÞ, ℝ n ðAÞ and ℓ 2 ðAÞ as examples of it have been introduced.In addition, fuzzy soft linear operator T in fuzzy soft Hilbert space H has been investigated.Moreover, the fuzzy soft orthogonal family and the fuzzy soft orthonormal family with examples on them have been established and some related theorems, examples and many more results of fuzzy soft linear operators with their proofs have been introduced.Furthermore, fuzzy soft right shift operator and fuzzy soft left shift operator with examples, on ℓ 2 ðAÞ, have been investigated.In addition, we have defined the fuzzy soft resolvent set, the fuzzy soft spectrum, the fuzzy soft spectral radius, the fuzzy soft point spectrum, the fuzzy soft approximate point spectrum, the fuzzy soft comparison spectrum, the fuzzy soft residual spectrum and the fuzzy soft continuous spectrum of the fuzzy soft linear operators and we have given the relations between them.Moreover, it has been shown that, on ℓ 2 ðAÞ, 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 j λj< 1.To make the picture complete, we have proved that the fuzzy soft Hilbert space has the fuzzy soft self-duality property.Our results can be considered as generalizations for the well-known (crisp) ones.This type of investigations fills some gaps in the literature.The authors can introduce new results by using similar techniques in this paper.Some special types of fuzzy soft linear operators in fuzzy soft Hilbert spaces will be studied in further investigations depending on the definition of the fuzzy soft linear operator in fuzzy soft Hilbert space given in this paper.Finally, there are still many topics to study by applying the fuzzy soft notion on them.

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 H be a fuzzy soft inner product space and T∈ Lð HÞ.Then, we have Rð TÞ ⊥ = Ñ ð T * Þ, where Rð TÞ is the fuzzy soft range of T and Ñ ð T * Þ is the fuzzy soft null (fuzzy soft kernel) of T * .

Table 2 :
[27]esentation of the fuzzy soft set ðG, AÞ.Definition 8 (The complement of a fuzzy soft point)[27].The fuzzy soft point ũf G C ðeÞ is called the fuzzy soft complement of a fuzzy soft point ũf GðeÞ , denoted by ðũ f GðeÞ Þ ⊆ E be the set of parameters.Then, ðu 2 , e 5 , G, 0:6Þ = ũ2 0:6 Gðe 5 Þ is a fuzzy soft point over U. C , if for the element e ∈ A and u ∈ U, Example 35.The fuzzy soft identity operator Ĩ : H ⟶ H defined by Ĩðṽ f GðeÞ Þ = ṽf GðeÞ , for all ṽf GðeÞ ∈ H, is a fuzzy soft linear operator on H. Example 36.The fuzzy soft zero operator 0 : H → H defined by 0ðṽ f GðeÞ Þ = θ, for all ṽf GðeÞ ∈ H, is a fuzzy soft linear operator on H.
and for all fuzzy soft scalars c1 , c2 .Definition 34 (Fuzzy soft linear operator in H).Let H be a fuzzy soft Hilbert space.A fuzzy soft linear operator T : Dð ⊂ HÞ ⟶ H is called a fuzzy soft linear operator in H .If D = H, then T is a fuzzy soft linear operator on H, written T∈ Lð HÞ .If T is fuzzy soft bounded (i.e., ∃ k∈ℝ + ðAÞ: kT ðṽ f GðeÞ Þk ≤k g kṽ f GðeÞ k ∀ ṽf GðeÞ ∈ H), then T∈ Bð HÞ.