# On Fuzzy Soft Linear Operators in Fuzzy Soft Hilbert Spaces

Authors: Nashat Faried; Mohamed S. S. Ali; Hanan H. Sakr
Journal: Abstract and Applied Analysis
 (2020)

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## 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 operatorT~ in the fuzzy soft Hilbert space H~ based on the definition of the fuzzy soft inner product space U~,·,·~ in terms of the fuzzy soft vector v~fGe modified in our work. Moreover, it is shown that ℂnA, ℝnA and ℓ2A 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 ℓ2A. In addition, it is proved, on ℓ2A, 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 λ~<~1~. Finally, it is shown that the fuzzy soft Hilbert space is fuzzy soft self-dual in this generalized setting.


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## Body

## 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 X is defined by its characteristic function from X to 0,1, a fuzzy set on a domain X is defined by its membership (characteristic) function from X to 0,1.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 integration, 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 [8–12]).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 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. Section2 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.


## 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].
LetU be a universal set (space of points or objects). A fuzzy set (class) X over U is a set characterized by a function fX:U→0,1. fX is called the membership, characteristic or indicator function of the fuzzy set X and the value fXu is called the grade of membership of u∈U in X.Definition 2 (Soft set) ([2, 26]).
A pairU,E 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 2U. A soft set is identified as a set of ordered pairs: U,E=e,Ue:e∈EandGe∈2U.Example 3.
Consider a soft setF,A 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=m1,m2,m3,m4,m5 and A=a1,a2,a3, where aii=1,2,3 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 F,A over U by the relation F,A=a1,m2,m4,a2,m5,a3,m1,m3,m4. This soft set can be represented in Table 1.Definition 4 (Fuzzy soft set) [13].
LetU be a universal set, E be a set of parameters and A⊆E. A pair G,A is called a fuzzy soft set over U, where G is a mapping given by G:A⟶FU, FU 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 0,1. The family of all fuzzy soft sets G,A over a universal set U, in which all the parameter sets A are the same, is denoted by FSSUA=FSSU~.Example 5.
Example (3) can be characterized by a membership function instead of 0,1 numbers which associates each element with a real number in the interval 0,1, then we can write the fuzzy soft set G,A over U as G,A=Ga1=m1,0.1,m2,0.6,m4,0.8,m5,0.3,Ga2=m4,0.2,m5,0.8,Ga3=m1,1,m3,0.7,m4,0.9. This fuzzy soft set can be represented in Table 2.Table 1
Representation of the soft setF,A.
Ua1a2a3m1001m2100m3001m4101m5010Table 2
Representation of the fuzzy soft setG,A.
Ua1a2a3m10.101m20.600m3000.7m40.80.20.9m50.30.80The 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 setG,A∈FSSU~ is called a fuzzy soft point over U, denoted by ufGe,A (briefly denoted by u~fGe), if for the element e∈A and u∈U (α∈0,1 is the value of the membership degree),
(1)fGeu=α,ifu=u0∈Uande=e0∈A,0,ifu∈U−u0ore∈A−e0.
The fuzzy soft point can be considered as the quadrupleu0,e0,G,α.Example 7.
Suppose that there are four points in the universal setU, i.e.,U=u1,u2,u3,u4 and let A=e1,e2,e3,e4,e5⊆E be the set of parameters. Then, u2,e5,G,0.6=u~20.6Ge5 is a fuzzy soft point over U.Definition 8 (The complement of a fuzzy soft point) [27].
The fuzzy soft pointu~fGCe is called the fuzzy soft complement of a fuzzy soft point u~fGe, denoted by u~fGeC, if for the element e∈A and u∈U,
(2)fGCeu=1−fGeu,ifu=u0∈Uande=e0∈A,0,ifu∈U−u0ore∈A−e0.Example 9.
The complement of the fuzzy soft point stated in Example (7) is the fuzzy soft point u2,e5,G,0.4=u~20.4Ge5.
The collection of all fuzzy soft points overU is denoted by FSPUA=FSPU~. ℝA denotes the set of all fuzzy soft real numbers and ℝ+A denotes the set of all non-negative fuzzy soft real numbers (such as r~, in symbol). ℂA denotes the set of all fuzzy soft complex numbers (such as c~, in symbol). Note that the fuzzy soft zero vector θ~=0~,0~,0~,0~ and the fuzzy soft unit vector j~=1~,1~,1~,1~.Definition 10 (Fuzzy soft vector space)([17]).
LetU be a vector space over a field KK=ℝ and the parameter set E be the set of all real numbers ℝ and A⊆E. The fuzzy soft set G,A∈FSSU~ is called a fuzzy soft vector over U, denoted by vfGe,A (briefly denoted by v~fGe), if there is exactly one e∈A such that fGev=α for some v∈U and fGe′v=0 for all e′∈A−e(α∈0,1 is the value of the membership degree). The set of all fuzzy soft vectors over Uis denoted by FSVUA=FSVU~. The set V~=~FSVU~ is said to be a fuzzy soft vector space or a fuzzy soft linear space of U over K if V~e is a vector subspace of U, for all e∈A. V~ is a fuzzy soft vector space according to the following two operations:
(1)
v~f1Ge11+v~f2Ge22=v1+v2~f1Ge1+f2Ge2, for all v~f1Ge11,v~f2Ge22∈~V~(2)
r~·v~fGe=rv~fGre, for all v~fGe∈~V~ and for all r~∈~ℝA.Example 11.
Consider the Euclideann− dimensional space ℝn over ℝ.
LetA=1,2,3,⋯,n be the set of parameters. Let V~:A⟶Fℝn be defined as follows:
V~i=v~fnGenn∈~ℝnA; ith coordinate of v~fnGenn is θ~}, i=1,2,⋯,n.
ThenV~ is a fuzzy soft vector space or a fuzzy soft linear space of ℝn over ℝ.
In addition, Letv~fnGenn be a fuzzy soft element of V~ as follows:
v~fnGenn=j~,j~,⋯,j~,θ~ith,j~,⋯,j~∈~ℝnA, i=1,2,⋯,n. Then v~fnGenn is a fuzzy soft vector of V~.Definition 12 (Fuzzy soft metric space) [16].
A fuzzy soft metric spaceU~,d~ is a fuzzy soft set U~ with a fuzzy soft real-valued function d~:FSPU~×FSPU~⟶ℝ+A satisfying the fuzzy soft metric conditions as the following:
(FSM1)d~u~f1Ge11,u~f2Ge22≥~0~, for all u~f1Ge11,u~f2Ge22∈~FSPU~, and d~u~f1Ge11,u~f2Ge22=~0~⟺u~f1Ge11=~u~f2Ge22.
(FSM2)d~u~f1Ge11,u~f2Ge22=~d~u~f2Ge22,u~f1Ge11, for all u~f1Ge11,u~f2Ge22∈~FSPU~.
(FSM3)d~u~f1Ge11,u~f2Ge22≤~d~u~f1Ge11,u~f3Ge33+d~u~f3Ge33,u~f2Ge22, for all u~f1Ge11,u~f2Ge22,u~f3Ge33∈~FSPU~.Definition 13 (Fuzzy soft normed space) [17].
LetFSVU~ be a fuzzy soft vector space. Then, a mapping ∥·∥~:FSVU~⟶ℝ+A is said to be a fuzzy soft norm on FSVU~if∥·∥~ satisfies the following conditions:
(FSN1)v~fGe~≥~0~, for all v~fGe∈~FSVU~, and v~fGe~=~0~⟺v~fGe=~θ~.
(FSN2)r~·v~fGe~=~r~v~fGe~, for all v~fGe∈~FSVU~ and for all fuzzy soft scalar r~.
(FSN3)v~f1Ge11+v~f2Ge22~≤~v~f1Ge11~+v~f2Ge22~, for all v~f1Ge11,v~f2Ge22∈~FSVU~.


## 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 self-duality of fuzzy soft Hilbert space is introduced.Definition 14 (Fuzzy soft inner product space) [28].
LetFSVU~ be a fuzzy soft vector space. Then, the mapping ·,·~:FSVU~×FSVU~⟶ℂA is said to be a complex fuzzy soft inner product (shortly, fuzzy soft inner product) on FSVU~if·,·~ satisfies the following axioms:
(FSI1)v~fGe,v~fGe~≥~0~, for all v~fGe∈~FSVU~, and v~fGe,v~fGe~=~0~⟺v~fGe=~θ~.
(FSI2)v~f1Ge11,v~f2Ge22~=~v~f2Ge22,v~f1Ge11~¯, for all v~f1Ge11,v~f2Ge22∈~FSVU~, where bar denotes the complex conjugate of the fuzzy soft complex number.
(FSI3)c~·v~f1Ge11,v~f2Ge22~=c~v~f1Ge11,v~f2Ge22~, for all v~f1Ge11,v~f2Ge22∈~FSVU~ and for all fuzzy soft scalar c~.
(FSI4)v~f1Ge11+v~f2Ge22,v~f3Ge33~=~v~f1Ge11,v~f3Ge33~+v~f2Ge22,v~f3Ge33~, for all v~f1Ge11,v~f2Ge22,v~f3Ge33∈~FSVU~.The fuzzy soft vector spaceFSVU~ 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 U~,·,·~.Definition 15 (Real fuzzy soft inner product space) [28].
If the mapping·,·~ in the above Definition (14) is replaced by ·,·~:FSVU~×FSVU~⟶ℝA, 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)):v~f1Ge11,v~f2Ge22~=~v~f2Ge22,v~f1Ge11~, for all v~f1Ge11,v~f2Ge22∈~FSVU~.Theorem 16 (Fuzzy soft Cauchy-Schwartz inequality) [28].
LetU~,·,·~ be a fuzzy soft inner product space, then for all v~f1Ge11,v~f2Ge22∈~FSVU~, we have
(3)v~f1Ge11,v~f2Ge22~2≤~v~f1Ge11,v~f1Ge11~v~f2Ge22,v~f2Ge22~.Theorem 17 (see [28]).
For allv~fGe∈~FSVU~, a fuzzy soft inner product space U~,·,·~ can be considered as a fuzzy soft normed space with v~fGe~=~v~fGe,v~fGe~.Example 18.
ℂnA 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:
(4)v~fGe,u~gGa~=~∑i=1nv~fiGeiiu~giGaii¯,
for allv~fGe,u~gGa∈~ℂnA.Solution.
The proof is straightforward by applying the conditions of the complex fuzzy soft inner product space stated in Definition (14).Example 19.
ℝnA 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:
(5)v~fGe,u~gGa~=~∑i=1nv~fiGeiiu~giGaii¯,
for allv~fGe,u~gGa∈~ℝnA.Solution.
The proof is straightforward by applying the conditions of the real fuzzy soft inner product space stated in Definition (15).Example 20.
ℓ2A the space of all fuzzy soft square-summable sequences is a complex fuzzy soft inner product space with the complex fuzzy soft inner product defined as follows:
(6)v~fGe,u~gGa~=~∑i=1∞v~fiGeiiu~giGaii¯,for all v~fGe,u~gGa∈~ℓ2A.Solution.
The proof is straightforward by applying the conditions of the complex fuzzy soft inner product space stated in Definition (14).Example 21.
C0~,1~A the space of all fuzzy soft complex-valued continuous functions on 0~,1~ is a complex fuzzy soft inner product space with the complex fuzzy soft inner product defined as follows:
(7)η~fGe,ξ~gGa~=~∫i=0~1~η~fiGeiix~ξ~giGaiix~¯dx~,for all η~fGe,ξ~gGa∈C0~,1~A.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].
LetU~,·,·~ be a complex fuzzy soft inner product space and let v~f1Ge11,v~f2Ge22∈~FSVU~. Then, we can write the fuzzy soft polarization identity in the following formula:
(8)v~f1Ge11,v~f2Ge22~=~1~4~v~f1Ge11+v~f2Ge22~2−v~f1Ge11−v~f2Ge22~2+i~v~f1Ge11+i~v~f2Ge22~2−i~v~f1Ge11−i~v~f2Ge22~2for all v~f1Ge11,v~f2Ge22∈~FSVU~.Definition 23 (Fuzzy soft Hilbert space) [29].
LetU~,·,·~ 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 H~,·,·~ (shortly H~). It is clear that every fuzzy soft Hilbert space is a fuzzy soft Banach space.Example 24.
The spaceℂnA 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:
(9)v~fGe,u~gGa~=~∑i=1nv~fiGeiiu~giGaii¯,for all v~fGe,u~gGa∈~ℂnA.Solution.
We haveℂnA is an fuzzy soft inner product space from Example (18). Using Theorem (17), we obtain:
(10)v~fGe~=~v~fGe,v~fGe~=~∑i=1nv~fiGeiiv~fiGeii¯=~∑i=1nv~fiGeii212,and since ℂnA is fuzzy soft complete in this fuzzy soft norm. Then, ℂnA is an fuzzy soft complete fuzzy soft inner product space, i.e., an fuzzy soft Hilbert space.Example 25.
The spaceℝnA 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:
(11)v~fGe,u~gGa~=~∑i=1nv~fiGeiiu~giGaii¯,for all v~fGe,u~gGa∈~ℝnA.Solution.
The proof is easy by using Example (19) and Theorem (17), similarly as Example (24).Example 26.
The spaceℓ2A 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:
(12)v~fGe,u~gGa~=~∑i=1∞v~fiGeiiu~giGaii¯,for all v~fGe,u~gGa∈~ℓ2A.Solution.
The spaceℓ2A is fuzzy soft complete as an fuzzy soft normed space. But ℓ2A is an fuzzy soft inner product space from Example (20). Then, ℓ2A is an fuzzy soft Hilbert space.Definition 27 (Fuzzy soft orthogonal family) [29].
LetH~ be a fuzzy soft inner product space. A family v~fiGeii of fuzzy soft elements of H~ is called a fuzzy soft orthogonal family if
(13)v~fiGeii⊥~v~fjGejj;i≠j,i.e.,v~fiGeii,v~fjGejj~=~0~;i≠j.Definition 28 (Fuzzy soft orthonormal family) [29].
LetH~ be a fuzzy soft inner product space. A family v~fiGeii of fuzzy soft elements of H~ is called a fuzzy soft orthonormal family if v~fiGeii is a fuzzy soft orthogonal family (i.e., satisfies the Condition (13)) and v~fiGeii~=~1~,∀i,i.e.,v~fiGeii,v~fiGeii~=~1~;i=j.Remark 29 (see [29]).
Note that ifv~fiGeii is a fuzzy soft orthonormal family, then v~fiGeii,v~fjGejj~=~δ~ij, where
(14)δ~ij=1~,ifi=j,0~,ifi≠j.Example 30.
The fuzzy soft set of fuzzy soft elementsj~,θ~,θ~, θ~,j~,θ~ and θ~,θ~,j~ is a fuzzy soft orthonormal family in ℝ3A.Solution.
Lete~1=~j~,θ~,θ~, e~2=~θ~,j~,θ~ and e~3=~θ~,θ~,j~. It is clear that e~i,e~j~=~δ~ij; i,j=1,2,3.Example 31.
The fuzzy soft set of fuzzy soft elementsj~,θ~,θ~,θ~,⋯,θ~,⋯, θ~,j~,θ~,θ~,⋯,θ~,⋯ and θ~,θ~,j~,θ~,θ~,⋯,θ~,⋯,⋯ is a fuzzy soft orthonormal family in ℓ2A.Solution.
Lete~1=~j~,θ~,θ~,θ~,⋯,θ~,⋯, e~2=~θ~,j~,θ~,θ~,⋯,θ~,⋯ and e~3=~θ~,θ~,j~,θ~,θ~,⋯,θ~,⋯. It is clear that e~i,e~j~=~δ~ij; i,j=1,2,3,⋯.Example 32.
LetH~=~C0~,1~A be the space of all fuzzy soft complex-valued continuous functions on 0~,1~. For k=0,±1,±2,⋯, let e~k⊂~H~ be defined by:
(15)e~kx~=~e~2πikx~,x~∈~0~,1~.
Then,e~k is a fuzzy soft orthonormal family in C0~,1~A.Solution.
(16)e~k,e~j~=~∫0~1~e~kx~e~jx~¯dx~=~∫0~1~e~2πikx~e~−2πijx~dx~=~∫0~1~e~2πik−jx~dx~=e~2πik−jx~0~1~2πik−j,ifk≠j∫0~1~dx~,ifk=j=e~2πik−j−1~2πik−j,ifk≠j∫0~1~dx~,ifk=j=0~,ifk≠j1~,ifk=j.
Hence,e~k, where e~k is defined as in (15), is a fuzzy soft orthonormal family in C0~,1~A.Definition 33 (Fuzzy soft linear operator).
LetT~:FSVU~1→FSVU~2 be an operator. Then T~ is said to be fuzzy soft linear if
(1)
T~v~f1Ge11+v~f2Ge22=~T~v~f1Ge11+T~v~f2Ge22, For all fuzzy soft elements v~f1Ge11,v~f2Ge22∈~FSVU~1 (additivity).(2)
T~c~v~fGe=~c~T~v~fGe, for all fuzzy soft element v~fGe∈~FSVU~1 and for all fuzzy soft scalar c~ (homogeneity).
The Conditions (1, 2) can be combined in one condition as follows:
T~c~1v~f1Ge11+c~2v~f2Ge22=~c~1T~v~f1Ge11+c~2T~v~f2Ge22, for all fuzzy soft elements v~f1Ge11,v~f2Ge22∈~FSVU~1 and for all fuzzy soft scalars c~1,c~2.Definition 34 (Fuzzy soft linear operator inH~).
LetH~ be a fuzzy soft Hilbert space. A fuzzy soft linear operator T~:D~⊂~H~⟶H~ is called a fuzzy soft linear operator in H~. IfD~=~H~, then T~ is a fuzzy soft linear operator on H~, written T~∈~L~H~. IfT~ is fuzzy soft bounded (i.e., ∃k~∈~ℝ+A: T~v~fGe~≤~k~v~fGe~∀v~fGe∈~H~), then T~∈~B~H~.Example 35.
The fuzzy soft identity operatorI~:H~⟶H~ defined by I~v~fGe=v~fGe, for all v~fGe∈~H~, is a fuzzy soft linear operator on H~.Example 36.
The fuzzy soft zero operator0~:H~→H~ defined by 0~v~fGe=θ~, for all v~fGe∈~H~, is a fuzzy soft linear operator on H~.Definition 37 (Fuzzy soft right shift operator).
LetH~ be a fuzzy soft Hilbert space. If v~fGe=~v~f1Ge11,v~f2Ge22,v~f3Ge33∈~H~. Define the fuzzy soft operator R~ as follows:
(17)R~v~fGe=~R~v~f1Ge11,v~f2Ge22,v~f3Ge33=~θ~,v~f1Ge11,v~f2Ge22.
The fuzzy soft operatorR~ is called the fuzzy soft right shift operator.Example 38.
The fuzzy soft right shift operatorR~:H~⟶H~ defined as above in Definition (37) is a fuzzy soft linear operator on H~.Definition 39 (Fuzzy soft left shift operator).
LetH~ be a fuzzy soft Hilbert space. If v~fGe=~v~f1Ge11,v~f2Ge22,v~f3Ge33∈~H~. Define the fuzzy soft operator L~ as follows:
(18)L~v~fGe=~L~v~f1Ge11,v~f2Ge22,v~f3Ge33=~v~f2Ge22,v~f3Ge33,θ~.
The fuzzy soft operatorL~ is called the fuzzy soft left shift operator.Example 40.
The fuzzy soft left shift operatorL~:H~⟶H~ defined as above in Definition 3.9 is a fuzzy soft linear operator on H~.Definition 41 (Fuzzy soft adjoint operator inH~).
The fuzzy soft adjoint operatorT~∗~ of a fuzzy soft linear operator T~ is defined by T~v~f1Ge11,v~f2Ge22~=~v~f1Ge11,T~∗~v~f2Ge22~, for all v~f1Ge11∈~H~.Example 42.
The fuzzy soft left shift operatorL~ defined in Definition (39) is the fuzzy soft adjoint of the fuzzy soft right shift operator R~ defined in Definition (37).Example 43.
LetH~=~ℓ2A be the space of all fuzzy soft square-summable sequences. If v~fGe=~v~f1Ge11,v~f2Ge22,v~f3Ge33,⋯∈~ℓ2A. 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) U~ in ℓ2A as follows:
(19)U~v~fGe=~U~v~f1Ge11,v~f2Ge22,v~f3Ge33,⋯=~θ~,v~f1Ge11,v~f2Ge22,v~f3Ge33,⋯.
Ife~n is a fuzzy soft orthonormal basis for ℓ2A, i.e., the set of fuzzy soft elements
(20)e~1=~j~,θ~,θ~,θ~,⋯,θ~,⋯,e~2=~θ~,j~,θ~,θ~,⋯,θ~,⋯,e~3=~θ~,θ~,j~,θ~,θ~,⋯,θ~,⋯,⋮is a fuzzy soft orthonormal family (fuzzy soft orthonormal basis) in ℓ2A. Then, by using (19), we get:
(21)U~e~n=~e~n+1,n=1,2,3,⋯.
For the fuzzy soft adjointU~∗~, we have:
(22)U~∗~e~i,e~j~=~e~i,U~e~j~.
Therefore, by using Formula (21) in (22), the following is obtained:
(23)U~∗~e~i,e~j~=~e~i,e~j+1~=~δ~i,j+1,i,j=1,2,⋯.
Since(24)U~∗~e~1,e~j~=~δ~1,j+1j+1>1=~0~,foralle~j,j=1,2,⋯,then
(25)U~∗~e~1=~θ~.
Ifi>1, then we have:
(26)δ~i,j+1=~δ~i−1,j=~e~i−1,e~j~.
Thus, by substituting from (26) in (23), we get:
(27)U~∗~e~i,e~j~=~e~i−1,e~j~and therefore, the following formula is obtained:
(28)U~∗~e~i=~e~i−1,i=2,3,⋯.
Now, by applying (25) and (28), we have:
(29)U~∗~v~fGe=~U~∗~v~f1Ge11,v~f2Ge22,v~f3Ge33,⋯=~U~∗~v~f1Ge11e~1+v~f2Ge22e~2+v~f3Ge33e~3+⋯=~v~f1Ge11U~∗~e~1+v~f2Ge22U~∗~e~2+v~f3Ge33U~∗~e~3+⋯=~v~f1Ge11×θ~+v~f2Ge22e~1+v~f3Ge33e~2+⋯.
Then, by substituting from (20) in (29), we obtain:
(30)U~∗~v~fGe=~v~f2Ge22j~,θ~,θ~,θ~,⋯,θ~,⋯+v~f3Ge33θ~,j~,θ~,θ~,⋯,θ~,⋯+⋯.=~v~f2Ge22,v~f3Ge33,⋯.
Hence, we finally get:(31)U~∗~v~fGe=~U~∗~v~f1Ge11,v~f2Ge22,v~f3Ge33,⋯=~v~f2Ge22,v~f3Ge33,⋯.
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.
LetH~ be a fuzzy soft inner product space and T~∈~L~H~. Then, we have R~T~⊥~=~N~T~∗~, where R~T~ is the fuzzy soft range of T~ and N~T~∗~ is the fuzzy soft null (fuzzy soft kernel) of T~∗~.Proof.
Letv~f1Ge11∈~H~ and v~f2Ge22∈~R~T~⊥~. Then, we get v~f1Ge11,T~∗~v~f2Ge22~=~T~v~f1Ge11,v~f2Ge22~=~0~, for all v~f1Ge11∈~H~. Therefore T~∗~v~f2Ge22=~θ~, and thus v~f2Ge22∈~N~T~∗~. That is to say that R~T~⊥~⊆~N~T~∗~. Conversely, let v~f2Ge22∈~N~T~∗~, then T~∗~v~f2Ge22=~θ~. Therefore, v~f1Ge11,T~∗~v~f2Ge22~=~0~ for all v~f1Ge11∈~H~. But we have that T~v~f1Ge11,v~f2Ge22~=~v~f1Ge11,T~∗~v~f2Ge22~ for all v~f1Ge11∈~H~. Then T~v~f1Ge11,v~f2Ge22~=~0~ for all v~f1Ge11∈~H~. But we know that T~v~f1Ge11∈~R~T~, then v~f2Ge22∈~R~T~⊥~. That is to say that N~T~∗~⊆~R~T~⊥~. Hence, R~T~⊥~=~N~T~∗~.Theorem 45.
LetT~,T~1,T~2∈~B~H~, where H~ is a fuzzy soft Hilbert space and let λ~∈~ℂA. Then, T~∗~∗~=~T~, λ~T~∗~=~λ~¯T~∗~, T~1+T~2∗~=~T~1∗~+T~2∗~, T~1T~2∗~=~T~2∗~T~1∗~, and if T~−1~ exists, then T~∗~−1~ exists and T~−1~∗~=~T~∗~−1~.Proof.
To prove the first result, letv~f1Ge11,v~f2Ge22∈~H~, then we have:
(32)T~v~f1Ge11,v~f2Ge22~=~v~f1Ge11,T~∗~v~f2Ge22~=~T~∗~v~f2Ge22,v~f1Ge11~¯=~v~f2Ge22,T~∗~∗~v~f1Ge11~¯=~T~∗~∗~v~f1Ge11,v~f2Ge22~.
To prove the last result, sincev~f1Ge11,I~∗~v~f2Ge22~=~I~v~f1Ge11,v~f2Ge22~=~v~f1Ge11,I~v~f2Ge22~. Then, I~∗~=~I~, where I~ is the fuzzy soft identity operator. Since T~−1~ exists, then T~T~−1~=~I~ and hence I~=~I~∗~=~T~T~−1~∗~=~T~−1~∗~T~∗~. Therefore, T~∗~−1~ exists and T~∗~−1~=~T~−1~∗~. The rest results can be proved similarly.Theorem 46.
LetT~∈~L~H~, where H~ is a fuzzy soft inner product space. IfT~v~fGe~=~v~fGe~ for all v~fGe∈~H~, then T~v~f1Ge11,T~v~f2Ge22~=~v~f1Ge11,v~f2Ge22~ for all v~f1Ge11,v~f2Ge22∈~H~.Proof.
By using Theorem (22), we have:
(33)T~v~f1Ge11,T~v~f2Ge22~=~1~4~T~v~f1Ge11+T~v~f2Ge22~2−T~v~f1Ge11−T~v~f2Ge22~2+i~T~v~f1Ge11+i~T~v~f2Ge22~2−i~T~v~f1Ge11−i~T~v~f2Ge22~2=~1~4~T~v~f1Ge11+v~f2Ge22~2−T~v~f1Ge11−v~f2Ge22~2+i~T~v~f1Ge11+i~v~f2Ge22~2−i~T~v~f1Ge11−i~v~f2Ge22~2.
Now, using the given thatT~v~fGe~=~v~fGe~ for all v~fGe∈~H~, we obtain that:
(34)T~v~f1Ge11,T~v~f2Ge22~=~1~4~v~f1Ge11+v~f2Ge22~2−v~f1Ge11−v~f2Ge22~2+i~v~f1Ge11+i~v~f2Ge22~2−i~v~f1Ge11−i~v~f2Ge22~2=~v~f1Ge11,v~f2Ge22~.Theorem 47.
LetT~∈~B~H~, where H~ is a fuzzy soft Hilbert space. Then, T~∗~ is fuzzy soft bounded and
(35)T~∗~~=~T~~.Proof.
Using Theorem (17) and Theorem (16), we have:
(36)T~∗~v~fGe~2=~T~∗~v~fGe,T~∗~v~fGe~=~T~T~∗~v~fGe,v~fGe~≤~T~T~∗~v~fGe~v~fGe~≤~T~~T~∗~v~fGe~v~fGe~,
for allv~fGe∈~H~. Then T~∗~v~fGe~≤~T~~v~fGe~ for all v~fGe∈~H~. Therefore,
(37)T~∗~~≤~T~~
and thusT~∗~ is fuzzy soft bounded. Similarly, using (1) from Theorem (45), we get:
(38)T~~=~T~∗~∗~~≤~T~∗~~.
Hence, from (37), (38), we obtain that T~∗~~=~T~~.Theorem 48.
LetT~∈~B~H~, where H~ is a fuzzy soft Hilbert space. Then, T~∗~T~~=~T~~2=~T~T~∗~~.Proof.
We have for allv~fGe∈~H~:
(39)T~∗~T~v~fGe~≤~T~∗~~T~v~fGe~≤~T~∗~~T~~v~fGe~.
By using (35) from Theorem (47) in (39), we obtain that T~∗~T~v~fGe~≤~T~~2v~fGe~ for all v~fGe∈~H~. Hence,
(40)T~∗~T~~≤~T~~2.
Conversely, using Theorem (17), we get for all v~fGe∈~H~:
(41)T~v~fGe~2=~T~v~fGe,T~v~fGe~=~v~fGe,T~∗~T~v~fGe~≤~T~∗~T~~v~fGe~2.
Then,T~v~fGe~≤~T~∗~T~~1/2v~fGe~ for all v~fGe∈~H~. Hence, T~~≤~T~∗~T~~1/2. That is to say that
(42)T~~2≤~T~∗~T~~.
Therefore, from (40), (42), the following result is obtained:
(43)T~∗~T~~=~T~~2.
Now, by applying this result (43) for T~∗~ in place of T~, we have T~T~∗~~=~T~∗~∗~T~∗~~=~T~∗~~2=~T~~2.Definition 49 (Fuzzy soft resolvent set ofT~).
LetT~∈~B~H~, where H~ is a fuzzy soft Hilbert space. Then, we define ρ~T~=~λ~∈~ℂA:λ~I~−T~−1~ exists and fuzzy soft bounded} =~λ~∈~ℂA:λ~>~T~~.
We callρ~T~ the fuzzy soft resolvent set of a fuzzy soft linear operator T~.Definition 50 (Fuzzy soft spectrum ofT~).
LetT~∈~B~H~, where H~ is a fuzzy soft Hilbert space. Then, we define σ~T~=~ρ~T~C=~λ~∈~ℂA:λ~≤~T~~. We call σ~T~ the fuzzy soft spectrum of a fuzzy soft linear operator T~.Definition 51 (Fuzzy soft spectral radius ofT~).
Since, forT~∈~B~H~, where H~ is a fuzzy soft Hilbert space, σ~T~ is non-empty fuzzy soft compact (fuzzy soft closed and fuzzy soft bounded), we define r~σ~T~=~supλ~:λ~∈~σ~T~. We call r~σ~T~ the fuzzy soft spectral radius of a fuzzy soft linear operator T~. Also, we can show that r~σ~T~=~limn→∞T~n~1/n.Definition 52 (Fuzzy soft point spectrum ofT~).
λ~∈~ℂA is said to be a fuzzy soft eigenvalue of a fuzzy soft linear operator T~ifλ~I~−T~ is not fuzzy soft injective, i.e., there exists a non-zero fuzzy soft element v~fGe∈~H~ such that T~v~fGe=~λ~v~fGe. Moreover, v~fGe≠~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 σ~pT~. We have σ~pT~⊂~σ~T~.Definition 53 (Fuzzy soft approximate point spectrum ofT~).
λ~∈~ℂA 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 v~fnGenn in H~ such that v~fnGenn~=~1~ and T~v~fnGenn−λ~v~fnGenn⟶0~. The set of all such λ~ is called the fuzzy soft approximate point spectrum of a fuzzy soft linear operator T~, denoted by σ~aT~. We have σ~aT~⊂~σ~T~.Definition 54 (Fuzzy soft comparison spectrum ofT~).
The set of allλ~∈~ℂA such that the fuzzy soft range of λ~I~−T~ is not fuzzy soft dense in H~, (i.e.R~λ~I~−T~¯≠~H~) is called the fuzzy soft comparison spectrum of a fuzzy soft linear operator T~, denoted by σ~ComT~.Definition 55 (Fuzzy soft residual spectrum ofT~).
The set of allλ~∈~ℂA such that λ~I~−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 σ~rT~. That is to say that:
(44)σ~rT~=~σ~ComT~\σ~pT~.Definition 56 (Fuzzy soft continuous spectrum ofT~).
The set of allλ~∈~ℂA such that λ~I~−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 σ~cT~. It is clear that σ~pT~∪~σ~cT~⊂~σ~aT~, and
(45)σ~T~=~σ~pT~∪~σ~cT~∪~σ~rT~,the terms on the right being mutually disjoint.Theorem 57.
LetH~ be a fuzzy soft Hilbert space and T~∈~B~H~. If λ~∈~σ~aT~, then, λ~≤~T~~.Proof.
Letλ~∈~σ~aT~, then there exists a fuzzy soft sequence of fuzzy soft elements v~fnGenn in H~ such that v~fnGenn~=~1~ and T~v~fnGenn→λ~v~fnGenn. Then,
T~v~fnGenn~⟶λ~v~fnGenn~=~λ~, i.e., limn→∞T~v~fnGenn~=~λ~. Therefore,
(46)λ~=~limn→∞T~v~fnGenn~≤~limnT~~v~fnGenn~=~T~~limnv~fnGenn~=~T~~.Corollary 58.
LetH~ be a fuzzy soft Hilbert space and T~∈~B~H~. Ifλ~∈~σ~pT~, then, λ~≤~T~~.Proof.
From (45), we get σ~pT~⊂~σ~aT~. Then, Theorem (57) completes the proof.Theorem 59.
IfT~∈~B~H~, where H~ is a fuzzy soft Hilbert space. Then,
(1)
T~ and T~∗~ have the same fuzzy soft resolvent set, i.e., ρ~T~=~ρ~T~∗~.(2)
T~ and T~∗~ have the same fuzzy soft spectrum, i.e., σ~T~=~σ~T~∗~.(3)
σ~rT~⊂~σ~pT~∗~ and σ~pT~⊂~σ~pT~∗~∪~σ~rT~.Theorem 60.
The fuzzy soft point spectrum of fuzzy soft right shift operator on fuzzy soft square-summable sequences spaceℓ2A has no fuzzy soft eigenvalues, i.e.,
(47)σ~pU~=~ϕ.Proof.
Letλ~∈~σ~pU~, then there exists θ~≠~v~fGe=~v~f1Ge11,v~f2Ge22,v~f3Ge33,⋯∈~ℓ2A such that U~v~fGe=~λ~v~fGe.
Therefore, by using (19), we obtain:
(48)θ~,v~f1Ge11,v~f2Ge22,v~f3Ge33,⋯=~λ~v~f1Ge11,λ~v~f2Ge22,λ~v~f3Ge33,⋯.
Then, from (48), we get that λ~v~f1Ge11=~θ~ and v~fnGenn=~λ~v~fn+1Gen+1n+1 for all n.
We have two cases:(1)
Ifλ~=~0~, then v~fnGenn=~θ~ for all n and hence v~fGe=~θ~.(2)
Ifλ~≠~0~, then v~f1Ge11=~θ~ and λ~v~f2Ge22=~θ~, then v~f2Ge22=~θ~, ⋯, and so on. Therefore, we obtain that v~fnGenn=~θ~ for all n and hence v~fGe=~θ~.
Now, each case of1,2 implies that v~fGe=~θ~, but this is a contradiction with v~fGe≠~θ~ assumed above at first. Hence, σ~pU~ must be empty, i.e., σ~pU~=~ϕ, which is required.Corollary 61.
The fuzzy soft residual spectrum of fuzzy soft right shift operator is equal to the fuzzy soft comparison spectrum of it on fuzzy soft square-summable sequences spaceℓ2A, i.e.,
(49)σ~rU~=~σ~ComU~.Proof.
The proof is straight forward by substituting from (47) in (44) and hence (49) is true.Theorem 62.
The fuzzy soft point spectrum of fuzzy soft left shift operator on fuzzy soft square-summable sequences spaceℓ2A is the fuzzy soft open disk ∣λ~∣<~1~, i.e.,(50)σ~pU~∗~=~λ~∈~ℂA:∣λ~∣<~1~.Proof.
Letλ~∈~σ~pU~∗~, then there exists θ~≠~v~fGe=~v~f1Ge11,v~f2Ge22,v~f3Ge33,⋯∈~ℓ2A such that U~∗~v~fGe=~λ~v~fGe. Therefore, by using (19), we obtain:
(51)v~f2Ge22,v~f3Ge33,v~f4Ge44,⋯=~λ~v~f1Ge11,λ~v~f2Ge22,λ~v~f3Ge33,⋯.
Then, from (51), we get that v~fn+1Gen+1n+1=~λ~v~fnGenn for all n. That is to say that,
(52)v~f2Ge22=~λ~v~f1Ge11
and(53)v~f3Ge33=~λ~v~f2Ge22.
Therefore, by substituting forv~f2Ge22 from (52) in (53), we have:
(54)v~f3Ge33=~λ~λ~v~f1Ge11=~λ~2v~f1Ge11.
In addition, we can repeat this process for(55)v~f4Ge44=~λ~v~f3Ge33
by substituting forv~f3Ge33 from (54) in (55) as follows:
(56)v~f4Ge44=~λ~λ~2v~f1Ge11=~λ~3v~f1Ge11⋮and so on. Thus, we get that:
(57)v~fn+1Gen+1n+1=~λ~nv~f1Ge11foralln,orv~fnGenn=~λ~n−1v~f1Ge11foralln.Now, we have two cases:(1)
Ifv~f1Ge11=~θ~, then v~fnGenn=~θ~ for all n and hence v~fGe=~θ~, which is a contradiction with v~fGe≠~θ~ assumed above at first.(2)
Ifv~f1Ge11≠~θ~, then ∑n=1∞v~fnGenn~2=~∑n=1∞λ~n−1v~f1Ge11~2=~∑n=1∞λ~2n−2v~f1Ge11~2, and therefore ∣λ~∣ must be less than 1~ for this fuzzy soft geometric series to be convergent. Hence, v~fnGenn~<~∞ if and only if ∣λ~∣<~1~.And finally this proves that (50) is satisfied.Theorem 63.
The fuzzy soft conjugate spaceH~∗~ of any fuzzy soft Hilbert space H~ is the same as H~, i.e.H~∗~=~H~ (H~ and H~∗~ are fuzzy soft isomorphic).Proof.
Firstly, we want to prove that every fuzzy soft elementw~hGe∈~H~ defines a fuzzy soft linear continuous functional f~∈~H~∗~ with the same fuzzy soft norm (i.e., to prove that f~~≤~w~hGe~). Let w~hGe∈~H~ and define f~∈~H~∗~ as follows f~v~fGa=~v~fGa,w~hGe~. To prove that f~ is fuzzy soft linear, let v~fGa,u~gGb∈~H~ and let λ~,μ~ be fuzzy soft scalars. Then, we have:
(58)f~λ~v~fGa+μ~u~gGb=~λ~v~fGa+μ~u~gGb,w~hGe~=~λ~v~fGa,w~hGe~+μ~u~gGb,w~hGe~=~λ~v~fGa,w~hGe~+μ~u~gGb,w~hGe~=~λ~f~v~fGa+μ~f~u~gGb.
Now, by using fuzzy soft Cauchy-Schwartz inequality, we get:(59)f~v~fGa=~v~fGa,w~hGe~≤~v~fGa~w~hGe~.
But, we have:(60)f~~=~supv~fGa≠~θ~f~v~fGav~fGa~.
Therefore, by substituting from (59) in (60), we obtain that f~~≤~supv~fGa≠~θ~v~fGa~w~hGe~/v~fGa~=~supv~fGa≠~θ~w~hGe~. That is to say that,
(61)f~~≤~w~hGe~.
Conversely, we want to prove that every fuzzy soft linear continuous functionalf~∈~H~∗~ defines a unique fuzzy soft element w~hGe∈~H~ with the same fuzzy soft norm (i.e., to prove that w~hGe~≤~f~~). Let f~∈~H~∗~ be a fuzzy soft linear continuous functional on H~, and let B~=~kerf~=~v~fGa∈~H~:f~v~fGa=~θ~. 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 v~f1Ga11,v~f2Ga22∈~B~ and let α~,β~ be fuzzy soft scalars. Then, f~v~f1Ga11=~θ~ and f~v~f2Ga22=~θ~. Thus, we obtain that
(62)f~α~v~f1Ga11+β~v~f2Ga22=~f~α~v~f1Ga11+f~β~v~f2Ga22=~α~f~v~f1Ga11+β~f~v~f2Ga22=~α~·θ~+β~·θ~=~θ~.
Therefore, we haveα~v~f1Ga11+β~v~f2Ga22∈~B~. Hence, B~ is a fuzzy soft linear manifold of H~. Second, to prove that B~ is fuzzy soft closed, let v~fnGann⟶v~fGa,v~fnGann∈~B~. Then, f~v~fnGann=~θ~. But f~ is fuzzy soft continuous, then f~v~fnGann⟶f~v~fGa. Thus, f~v~fGa=~θ~. Therefore, v~fGa∈~B~ and hence B~ is fuzzy soft closed. That is to say that, B~ is a fuzzy soft subspace of H~. Then, H~=~B~⊕~B~⊥~. Therefore, every fuzzy soft element v~fGa∈~H~ can be written uniquely as v~fGa=~u~gGb+z~qGd,u~gGb∈~B~,z~qGd⊥~B~. Then, f~u~gGb=~θ~, but f~z~qGd≠~θ~. Let v~fGa∉~B~ (i.e., v~fGa∈~H~−B~), then v~fGa=~u~gGb+z~qGd, where z~qGd≠~θ~ and f~z~qGd=~η~≠~0~. Let v~f1Ga11=~z~qGd/η~. Therefore,
(63)f~v~f1Ga11=~f~z~qGdη~=~η~η~=~1~.
Ifv~fGa is any fuzzy soft point of H~, then
(64)f~v~fGa=~ξ~=~ξ~·1~.
Thus, by substituting from (63) in (64), we obtain that f~v~fGa=~ξ~f~v~f1Ga11. Therefore, f~v~fGa−ξ~v~f1Ga11=~θ~. Then, v~fGa−ξ~v~f1Ga11∈~B~. Let v~fGa−ξ~v~f1Ga11=~u~gGb, u~gGb∈~B~. Hence, v~fGa=~u~gGb+ξ~v~f1Ga11, u~gGb∈~B~, ξ~v~f1Ga11⊥~B~. Therefore, we have:
(65)v~fGa,v~f1Ga11~=~u~gGb+ξ~v~f1Ga11,v~f1Ga11~=~u~gGb,v~f1Ga11~+ξ~v~f1Ga11,v~f1Ga11~=~0~+ξ~v~f1Ga11,v~f1Ga11~=~ξ~v~f1Ga11~2.
Then,v~fGa,v~f1Ga11/v~f1Ga11~2~=~ξ~=~f~v~fGa. Let w~hGe=~v~f1Ga11/v~f1Ga11~2∈~H~, then f~~=~supv~fGa≠~θ~f~v~fGa/v~fGa~≥~f~w~hGe/w~hGe~=~w~hGe,w~hGe~/w~hGe~=~w~hGe~w~hGe~/w~hGe~=~w~hGe~. That is to say that,
(66)w~hGe~≤~f~~.
Hence, from (61) and (66), we obtain that f~~=~w~hGe~.
At the end, we have to show that the fuzzy soft elementw~hGe is unique. Suppose that
(67)f~v~fGa=~v~fGa,w~`hGe``~,∀v~fGa∈~H~.⟹v~fGa,w~hGe~=~v~fGa,w~`hGe``~,∀v~fGa∈~H~.⟹v~fGa,w~hGe−w~`hGe``~=~0~,∀v~fGa∈~H~.⟹w~hGe−w~`hGe``=~θ~.⟹w~hGe=~w~`hGe``.


## 4. 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 spaceU~,·,·~ by using the concept of fuzzy soft vector v~fGe modified in our work and have shown that ℂnA, ℝnA, ℓ2A and C0~,1~A are suitable examples of fuzzy soft inner product spaces. Also, the definition of the fuzzy soft Hilbert space H~ with ℂnA, ℝnA and ℓ2A 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 ℓ2A, 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 ℓ2A, 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 λ~<~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.



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