Properties of Fuzzy Compact Linear Operators on Fuzzy Normed Spaces

In this paper the definition of fuzzy normed space is recalled and its basic properties. Then the definition of fuzzy compact operator from fuzzy normed space into another fuzzy normed space is introduced after that the proof of an operator is fuzzy compact if and only if the image of any fuzzy bounded sequence contains a convergent subsequence is given. At this point the basic properties of the vector space FC(V,U)of all fuzzy compact linear operators are investigated such as when U is complete and the sequence (Tn) of fuzzy compact operators converges to an operator T then T must be fuzzy compact. Furthermore we see that when T is a fuzzy compact operator and S is a fuzzy bounded operator then the composition TS and ST are fuzzy compact operators. Finally, if T belongs to FC(V,U) and dimension of V is finite then T is fuzzy compact is proved.


Introduction:
Some results of fuzzy complete fuzzy normed spaces were studied by Saadati and Vaezpour in 2005 (1).Properties of fuzzy bounded linear operators on a fuzzy normed space were investigated by Bag and Samanta in 2005 (2).The fuzzy normed linear space and its fuzzy topological structure were studied by Sadeqi and Kia in 2009 (3).Properties of fuzzy continuous operators on a fuzzy normed linear spaces were studied by Nadaban in 2015 (4).The definition of the fuzzy norm of a fuzzy bounded linear operator was introduced by Kider and Kadhum in 2017 (5).Fuzzy functional analysis is developed by the concepts of fuzzy norm and a large number of researches by different authors have been published for reference please see (6,7,8,9,10).The structure of this paper is as follows: In section two we recall the definition of fuzzy normed space (11) also some basic definitions and properties of this space, that we will need later in this paper and the definition of three types of fuzzy convergence sequence of operators.The main results can be found in third section.The aim of this paper is to introduce the notion of fuzzy compact operator from a fuzzy normed space to another fuzzy normed space and some basic properties of this type of operators are investigated and proved.
Assume that (V, L V , ⨂ )is a fuzzy normed space and let a ∈ V , t > 0 , 0 <  < 1.If L V (a, t) > (1 − q) then there is s with 0 < s <  such that L V (a, s) > (1 − q).Definition 6: (5) Suppose that (V, L V , ⨂)is a fuzzy normed space.Put FB(a ,p, t) = {b∈V: Assume that ( V, L V , ⨂)is a fuzzy normed space.A ⊆V is called fuzzy bounded if we can find t > 0 and 0 <  < 1 such that L V (a, t) > (1 − q) for each a ∈ A.

Definition 8 :(5)
A sequence(a n ) in a fuzzy normed space(V, L V , * ) is called converges to a ∈ V if for each q > 0 and t > 0 we can find N∈ ℕ with L V [a n − a, t] > (1 − q) for all  ≥ .This is equivalent to lim n→∞ L V [a n − a, t] = 1.Or in other word lim n→∞ a n = a or simply represented by a n → a, the vector a is known as the limit of (a n ).

Definition 9 :(5)
A sequence(a n )in a fuzzy normed space (V, L V , ⨂)is said to be a Cauchy sequence if for all 0 <  < 1 ,  > 0 there is a number N∈ ℕ with L V [a m − a n , t] > (1 − q) for all m, n≥ N. Definition 10:(5) Suppose that (V, L V , ⨂)is a fuzzy normed space and let A be a subset of V. Then A is said to be open if for each a in A there is FB(a, p, t) such that FB(a, p, t) ⊆ A. Also a subset B is said to be closed if B c is an open set in V.

Definition 11:(5)
Suppose that (V, L V , ⨂)is a fuzzy normed space and let A be a subset of V.Then, the closure ofA is written byA ̅ or CL(A)and which is A ̅ = ⋂{B⊆V:B is closed and A⊆B} Definition 12: (5) Suppose that (V, L V , ⨂) is a fuzzy normed space and A ⊆ V. Then A is called dense in V when A ̅ = V.Lemma 13: (5) Assume that (V, L V , ⨂) is a fuzzy normed space and A is a subset of V.Then, y ∈ A ̅ if and only if there is a sequence(y n ) in A with (y n ) converges to y.

Lemma 14:
If A and B are subsets of a fuzzy normed space (V, L V , ⨂) then A + B ̅̅̅̅̅̅̅ = A ̅ + B ̅ .Proof: Let a+b ∈ A ̅ + B ̅ then by Lemma 2.13 there is a sequence (a n ) in A such that lim n→∞ L V ( a n − a, t)=1 and there is a sequence (b n ) in B such that lim n→∞ L V ( b n − b, s)=1 for all t, s>0.Now Suppose that(V, L V , ⨂) is a fuzzy normed space and A is a subset of V. Then A is dense in Vif and only if for every  ∈ there is  ∈  such thatL V [x − a , t] > (1 − ε)for some 0 <  < 1 and  > 0.

Definition 16:(1)
A fuzzy normed space (V, L V , ⨂) is said to be complete if every Cauchy sequence in V converges to a point in V.

Definition 17:(2)
Suppose that (V, L V , ⨂) and(W, L W , ⨀) are two fuzzy normed spaces .The operator S: V → W is said to be fuzzy continuous at   ∈ V if for all t > 0 and for all 0 <  < 1there is s and there is β with, L V [v − v 0 , s] > (1 − β) we have L W [S(v) − S(v 0 ), t] > (1 − α) for all v ∈V.Theorem 18: (5) Suppose that (V, L V , ⨂)and(U, L U , ⨀)are two fuzzy normedspaces.The operator T: V → U is fuzzy continuous at a ∈V if and only if a n →a in V implies T(a n ) →T(a) in U. Definition 19: (5) Suppose that (V, L V , ⨂) and(W, L W , ⨀) are two fuzzy normed spaces.An operator T: D(T) →W is said to be fuzzy bounded if there exists r, 0 < r < 1 such that L W (Tx , t) ≥ (1 − r)⨂L V (x, t), for each x ∈ D(T) ⊆ X and t> 0 where ⨂ is a continuous tnorm and D(T) is the domain of T. Theorem 20: (5) Suppose that (V, L V , ⨂) and(W, L W , ⨀) are two fuzzy normed spaces.The operator S:D(S) →W is fuzzy bounded if and only if S(A) is fuzzy bounded for every fuzzy bounded subset A of D(S).
Put FB(V,W) ={S:V→W, S is a fuzzy bounded operator} when (V, L V , ⨂) and (W, L W , ⨀) are two fuzzy normed spaces (5).Theorem 21: (5) Suppose that (V, L V , ⨂) and(W, L W , ⨀)are two fuzzy normed spaces.Define L(T, t) =  ∈()   (, ) for all T ∈ FB(V, W)andt > 0 then (FB(V, W), L, * ) is fuzzy normed space.Theorem 22 : (5) Suppose that (V, L V , ⨂) and(W, L W , ⨀) are two fuzzy normed spaces with S:D(S) →W is a linear operator where D(S) ⊆V.Then, S is fuzzy bounded if and only if S is fuzzy continuous.Corollary 23 : (5) Suppose that (V, L V , ⨂) and(W, L W , ⨀) are two fuzzy normed spaces.Assume thatT: D(T) →W is a linear operator where D(T) ⊆V.Then, T is a fuzzy continuous if T is a fuzzy continuous atx ∈ D(T).Definition 24: (6) Suppose that (V, L V , ⨂) is a fuzzy normed space and W ⊆ V then, it is said to be totally fuzzy bounded if for any σ ∈ (0, 1), t > 0 we can find W  ={a 1 , a 2 , … , a  }in W with any v ∈ V there is some k A i then, Ψ is known as a finite sub covering of W.

Definition 26:(11)
A fuzzy normed space Theorem 27: (11) The fuzzy normed space(V, L V ⨂) is compact if and only if every (v n ) in V contains (v n k ) with v n k →v.

Lemma 28:
If A and B are two compact subsets of a fuzzy normed space(V, L V , ⨂) then, A+B is compact.Proof: Let { G i : i∈I} be an open covering for A+B then there are I 1 ⊆I and I 2 ⊆I such that{ G i : i∈ I 1 } is an open covering for A and { G k : k∈ I 2 } is an open covering for B. But A and B are compact so, there is a finite sub covering Ψ 1 of { G i : i∈ I 1 } for A and a finite sub covering Ψ 2 of { G k : k∈ I 2 } for B. Hence Ψ 1 ∪ Ψ 2 is a finite sub covering of { G i : i∈I } for A+B.Hence, A+B is compact.
Proposition 29: (6) Let (V, L V , ⨂) be a fuzzy normed space if V is totally fuzzy bounded then, V is fuzzy bounded.Proposition 30: (6) If the fuzzy normed space (V, L V , ⨂) is compact then, it is totally fuzzy bounded.

Definition 31 :(5)
A linear functional f from a fuzzy normed space(V, L V , ⨂) into the fuzzy normed space (F, L F , * ) is said to be fuzzy bounded if there exists r, 0 <  < 1 such that L F [f(x), t] ≥ (1 − r)⨂L V [x, t] for all x ∈ D(f) and t > 0. Furthermore, the fuzzy norm of f is and L F (f(x), t) ≥ L(f, t)⨂L V (x, t).Definition 32 : (5) Suppose that(V, L V , ⨂) is a fuzzy normed space.Then, the vector space FB(V, ) ={f:V→ , f is fuzzy bounded linear function } with a fuzzy norm defined by (, ) =    ((), ) form a fuzzy normed space which is called the fuzzy dual space of V. Definition 33:(5) A sequence(v n ) in a fuzzy normed space (V, L V , ⨂) is said to fuzzy weakly convergent if we can find v ∈ V with every h ∈ FB(V,R) lim n→∞ h(v n ) = h(v).This is writtenv n → w v the element v is said to be the weak limit to (v n ) and (v n ) is said to be fuzzy converges weakly to v. Definition 34: (6) Suppose that (V, L V , ⨂)and (U, L U , ⨀) are two fuzzy normed spaces.A sequence (T n )operators T n ∈ FB(V, U) is said to be 1.Fuzzy uniformly operator convergent if there is T: V → U withL[T n -T, t] → 1 for any  > 0 and n≥N.2.Fuzzy strong operator convergent if(T n v)converges in U for every v∈ V that is that is there is T: V → U withL U [T n v-Tv, t] → 1 for every  > 0 and n ≥ N. 3.Fuzzy weakly operator convergent if for every v ∈ V there is T: V → U withL R [f(T n v) − f(Tv), t] for every  > 0, f ∈ FB(U, ℝ) and n≥N.Definition 35: (6) Let (V, L V , ⨂) be a fuzzy normed space.A sequence(h n )offunctional h n ∈ FB(V, ℝ) is called 1) Fuzzy strong converges in the fuzzy norm onFB(V, R) that is there is h ∈ FB(V, ℝ) withL[h n − h, t] → 1 for all t > 0 this writtenh n → h 2) Fuzzy weak converges in the fuzzy norm on R that is there is h ∈ FB(V,ℝ) with h n (v) → f(v) for every v ∈ V written by lim n→∞ h n (v) = h(v).
and FB[a, p, t] are called open and closed fuzzy ball with the center ain V and radius p, with  > 0. Definition 7:(6)