Fuzzy compact linear operator 1

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1 Advances in Fuzzy Mathematics (AFM) X Volume 12, Number 2 (2017), pp Research India Publications Fuzzy compact linear operator 1 S. Chatterjee, T. Bag, S.K. Samanta Department of Mathematics, Visva-Bharati, Santiniketan West Bengal, India. tarapadavb@gmail.com Abstract In this paper, we introduce fuzzy compact linear operator on fuzzy normed linear space( with general t-norm setting ) and studied some of its basic properties. AMS subject classification: 03E70, 15A03. Keywords: Fuzzy normed linear space, Fuzzy compact linear operator. 1. Introduction In 1986, A.K. Katsaras [7] first introduced the idea of a fuzzy norm in a linear space. After that, several authors introduced the idea of fuzzy norm on linear space in different approaches (please see [1], [8], [10]). Following the definition of fuzzy normed linear space given by Bag and Samanta [1], the concept of fuzzy compact linear operators introduced by F. Lael et al. [6]. On the other hand M. Saheli et al. [11] introduced the concept of fuzzy compact linear operators on Felbin type s [8] fuzzy normed linear space. In this paper, we consider fuzzy normed linear space (X, N, ), (where is a general t-norm) introduced by Bag and Samanta in 2012 [3] and give the definition of fuzzy compact linear operator in such space. Some basic properties of compact linear operator are studied in fuzzy setting. The organization of the paper is as follows: Section 2 comprises some preliminary results. In section 3, a definition of fuzzy compact linear operators on fuzzy normed linear space is given, some examples of this type of compact linear operators are established, Some basic general properties of fuzzy linear compact operators are investigated. 1 This work is partially supported by the Special Assistance Programme (SAP) of UGC, New Delhi, India [Grant No. F 510/3/DRS-III/2015 (SAP- I)].

2 216 S. Chatterjee, T. Bag, S.K. Samanta 2. Preliminaries Definition 2.1. [2] A binary operation :[0, 1] [0, 1] [0, 1] is a t-norm if it satisfies the following conditions: (1) is associative and commutative; (2) a 1 = a a [0, 1]; (3) a b c d whenever a c and b d for each a, b, c, d [0, 1]. Definition 2.2. [3] Let U be a linear space over the field F (C or R). A fuzzy subset N of U R (R-the set of all real numbers) is called a fuzzy norm on U if (N1) t R with t 0, N(x, t) = 0; (N2) ( t R, t>0, N(x, t) = 1) iff x= 0; t (N3) t R, t>0, N(cx,t)= N(x, )if c = 0; c (N4) s, t R; x,u U; N(x + u, s + t) N(x,s) N(u, t); (N5) N(x,.) is a non-decreasing function of R and lim N(x,t) = 1. t The triplet (U, N, ) will be referred to as a fuzzy normed linear space. Remark 2.3. [4] From (N2) and (N4), it follows that N(x,.) is a non-decreasing function of R. So, we take modified form of (N5) by deleting the condition N(x,.) is non-decreasing. Assume that [1], (N6) t >0, N(x,t)>0 implies x = 0. Definition 2.4. [1] Let (U, N) be a fuzzy normed linear space. Let {x n } be a sequence in U. Then {x n } is said to be convergent if x U such that lim N(x n x,t) = 1 t >0. n In that case x is called the limit of the sequence {x n } and is denoted by lim x n. Definition 2.5. [1] Let (U, N) be a fuzzy normed linear space. A subset F of U is said to be closed if for any sequence {x n } in F converges to x. i.e. lim N(x n x,t) = 1 t >0 n implies that x F. Definition 2.6. [1] Let (U, N) be a fuzzy normed linear space. A subset B of U is said to be the closure of F if for any x B, a sequence {x n } in F such that lim N(x n x,t) = 1 t >0. We denote the set B by F. n Note. In this paper we denote f F instead of F.

3 Fuzzy compact linear operator 217 Definition 2.7. [1] Let (U, N) be a fuzzy normed linear space. A subset A of U is said to be compact if any sequence {x n } in A has a subsequence converging to an element of A. Definition 2.8. [1] A subset A of a fuzzy normed linear space is said to be fuzzy bounded if for each r, 0 <r<1 t>0 such that N(x,t) > 1 r x A. Note. Through out the paper we write bounded set instead of fuzzy bounded set. Definition 2.9. [4] Let (U, N, ) be a fuzzy normed linear space. B(x, α, t) is a set defined by B(x, α, t) ={y : N(x y,t) > 1 α}, α (0, 1). This set B(x, α, t) is denoted by open ball in (U, N, ). Theorem [3] Let (U, N) be a finite dimensional fuzzy normed linear space in which the underlying t-norm is continuous at (1, 1). Then a subset A is compact iff A is closed and bounded. Lemma [3] Let (U, N, ) be a fuzzy normed linear space satisfying (N6) and t- norm is continuous at (1, 1). If {x 1,x 2,...,x n } is a linearly independent set of vectors in X, then for each α (0, 1), c α > 0 such that for any set of scalars {β 1,β 2,...,β n }; {t >0 : N(x 1 β 1 + x 2 β 2 + +x n β n,t) 1 α} c α n i=1 β i. Definition [5] Let T : (X, N 1, 1 ) (Y, N 2, 2 ) be a linear operator where (X, N 1, 1 ) and (Y, N 2, 2 ) are fuzzy normed linear spaces. T is said to be fuzzy bounded if for each α (0, 1), M α > 0 such that t N 1 (x, ) 1 α N 2 (T x,s) α s >t, t >0. (2.14.1) M α Proposition [5] Let T : (X, N 1, 1 ) (Y, N 2, 2 ) be a linear operator where (X, N 1, 1 ) and (Y, N 2, 2 ) are fuzzy normed linear spaces. If T is fuzzy bounded then the relation (2.12.1) is equivalent to the relation {t >0 : N 2 (T x, t) α} M α ( {t >0 : N 1 (x, t) 1 α}) x X (2.15.1) Note [5]. The collection of all linear operators defined from a fuzzy normed linear space (X, N 1, 1 ) to another normed linear space (Y, N 2, 2 ) is denoted by L(X, Y ) and for the collection of fuzzy bounded linear operators is denoted by BF (X, Y ). Lemma Let (X, N, ) be a fuzzy normed linear space and the underlying t-norm be lower semicontinuous. Then for each α (0, 1), β α such that {t >0 : N(x+y,t) α} {t>0 : N(x,t) β}+ {t >0 : N(x,t) β} x,y X.

4 218 S. Chatterjee, T. Bag, S.K. Samanta Definition [5] An operator T : (X 1,N 1 ) (X 2,N 2 ) is said to be fuzzy continuous at x X if for every sequence {x n } in X 1 with x n x implies T(x n ) T(x), i.e., lim N 1(x n x,t) = 1 t>0implies lim N 2(T (x n ) T(x),t)= 1 t>0. n n Theorem [5]Let T : (X, N 1 ) (Y, N 2 ) be a linear operator where (X 1,N 1 ) and (X 2,N 2 ) are fuzzy normed linear spaces. If T is fuzzy bounded then it is fuzzy continuous. 3. Some results on fuzzy normed linear space In this section some results on fuzzy normed linear space are studied. Definition 3.1. Let (X, N, ) be a fuzzy normed linear space. The point x X is said to be a limit point of a set A X if for every α (0, 1) and t>0, B(x,α,t) A contains a point other than x. The collection of all limit points of A is called the derived set of A and is denoted by A. Lemma 3.2. Let (X, N, ) be a fuzzy normed linear space and M X. Then f M = M M. Proof. Let x M M. Case I: If x M. Then we choose a sequence {x n } with x n = x n N, and lim N(x n x,t) = lim N(θ,t) = 1, t>0. n n x f M. Case II: When x M. Suppose t > 0 be given. Then for each α n (0, 1) with lim n α n = 0, an open ball B(x,α n,t ) of x such that B(x,α n,t ) M contains a point other than x. Then y n B(x,α n,t ) M, n N. N(y n x,t )>1 α n, n N. lim n N(y n x,t ) = 1. Since t > 0 be arbitrary so lim n N(y n x,t) = 1, t >0. x f M. M M f M Again let x f M. Then a sequence {x n } in M such that, lim n N(x n x,t) = 1, t>0.

5 Fuzzy compact linear operator 219 So for each α (0, 1), and t>0 N N such that N(x n x,t) > 1 α, n N. Now, N(x N x,t) > 1 α and x N M B(x, α, t) M contains a point other than x, α (0, 1) andt>0. x M x M M Thus f M M M. f M = M M. Lemma 3.3. Let (X, N, ) be a fuzzy normed linear space and M X. Then x f M iff for a given ɛ>0 and 0 <α<1, y M such that N(x y,ɛ) > 1 α. Proof. Let x f M. Now by Lemma??, x M M. Case I: If x M then choose y = x and we get N(x y,ɛ) = N(θ,ɛ) = 1 > 1 α, for any ɛ>0and 0 <α<1. Case II: Let x/ M, then x M. So, x is a limit point of M. Thus for a given ɛ>0and 0 <α<1, an open ball B(x, α, ɛ) such that B(x, α, ɛ) M contains a point other than x. y M such that y B(x, α, ɛ) N(x y,ɛ) > 1 α Conversely, let x X and suppose that for a given ɛ>0and 0 <α<1, y M such that N(x y,ɛ) > 1 α. y B(x, α, ɛ) y B(x, α, ɛ) M B(x, α, ɛ) M contains a point other than x. x is a limit point of M. x M x M M x f M, (by Lemma 3.2). Lemma 3.4. Let (X, N, ) be a fuzzy normed linear space and A be a subset of X. Then for each sequence {x n } in A, a sequence {y n } in A such that, lim N(x n y n,t)= 1 t>0. n Proof. Let {x n } be a sequence in A and let t > 0 be given. Then for a sequence {ɛ n } with ɛ n 0asn, a sequence {y n } in A such that N(x n y n,t )>1 ɛ n (by Lemma 3.3). lim N(x n y n,t ) 1 lim ɛ n n n lim n N(x n y n,t ) = 1.

6 220 S. Chatterjee, T. Bag, S.K. Samanta Since t > 0 is arbitrary, lim N(x n y n,t)= 1 t>0. n Lemma 3.5. Let (X, N, ) be a fuzzy normed linear space and be lower semicontinuous. If A X is bounded then f A is also so. Proof. Let β (0, 1). By lower semicontinuity of, α 0 (0, 1) such that (1 α 0 ) (1 α 0 )>1 β. Now let x f A. Then a sequence {x n } in A such that lim N(x n x,t) = 1 t>0 (1) n Since A is bounded so for α 0 (0, 1), t 0 > 0 such that N(x n,t 0 )>1 α 0 n N (2) Again from (1), N(x n x,t 0 )>1 α 0 n n 0 (α 0,t 0 ) Now N(x,2t 0 ) N(x n0 x,t 0 ) N(x n0,t 0 ) >(1 α 0 ) (1 α 0 )>1 β N(x,t )>1 β, where 2t 0 = t > 0. Since β (0, 1) is arbitrary, thus for each β (0, 1), t (β) > 0 such that, N(x,t )>1 β f A is bounded. Theorem 3.6. Let (X, N, ) be a finite dimensional fuzzy normed linear space satifying (N6) and the underlying t norm be lower semicontinuous. Then every linear operator on X is bounded. Proof. Let dim X = n and {e 1,e 2,...,e n } be a basis for X. We take any x = n i=1 β i e i and consider a linear operator T on X. Since T is linear then T(x)= n i=1 β i T(e i ). Now by Lemma 2.17, for each α (0, 1), β α such that, {t>0; N 2 (β 1 T(e 1 ) + β 2 T(e 2 ) + +β n T(e n ), t) α} β 1 ( {t >0; N 2 (T (e 1 ), t) β}) + β 2 ( {t >0; N 2 (T (e 2 ), t) β}) + + β n ( {t >0; N 2 (T (e n ), t) β}) (1)

7 Fuzzy compact linear operator 221 Let Now from (1), p(α) = max 1 i n {t>0; N 2(T (e i ), t) β} {t >0; N 2 (β 1 T(e 1 ) + β 2 T(e 2 ) + +β n T(e n ), t) α} p n i=1 β i (2) By Lemma 2.13, for any α (0, 1) C α > 0 such that, {t >0; N 1 (β 1 e 1 + β 2 e 2 + +β n e n,t) 1 α} C α n i=1 β i Thus for β(α) (0, 1), C β (α) > 0 such that, Putting this in (2), {t >0; N 1 (β 1 e 1 + β 2 e 2 + +β n e n,t) 1 β} C β n i=1 β i {t>0; N 2 (β 1 T(e 1 ) + β 2 T(e 2 ) + +β n T(e n ), t) α} p C β ( {t >0; N 1 (β 1 e 1 + β 2 e 2 + +β n e n,t) 1 β}) (3) Since β α thus 1 β 1 α. So, {t >0; N 1 (x, t) 1 α} {t>0; N 1 (x, t) 1 β} {t>0; N 1 (x, t) 1 α} {t>0; N 1 (x, t) 1 β} So from (3) we get, {t>0; N 2 (β 1 T(e 1 ) + β 2 T(e 2 ) + +β n T(e n ), t) α} p ( {t >0; N 1 (β C 1 e 1 + β 2 e 2 + +β n e n,t) 1 α}) β Thus, {t>0; N 2 (T x, t) α} M α ( {t >0; N 1 (x, t) 1 α}) x X, where T is a fuzzy bounded linear operator. M α = p(α) C β (α)

8 222 S. Chatterjee, T. Bag, S.K. Samanta 4. Fuzzy Compact Linear Operator In this section a definition of fuzzy compact linear operator is given and some of its basic properties are studied. Definition 4.1. Let (X, N 1, 1 ) and (Y, N 2, 2 ) be two fuzzy normed linear space. A linear operator T : (X, N 1, 1 ) (Y, N 2, 2 ) is called a fuzzy compact linear operator if for every bounded subset M of X the subset T(M) of Y is relatively compact, i.e. f T(M)is a compact set. Example 4.2. Let X = Y = R 2 and (x, y) i = x + y, (x, y) R 2,i= 1, 2 and T : (X, 1 ) (Y, 2 ) be a compact linear operator. Define N i : X R [0, 1] by t N i (x, t) = 2 (t + x 1 )(t + x 2 ),t>0 0, t 0 for i = 1, 2, x= (x 1,x 2 ). Then (X, N 1, 1 ) and (Y, N 2, 2 ) are fuzzy normed linear space w.r.t. the continuous t-norm 1 = 2 = product, (Example 3.2 [3]). Let B be a bounded set w.r.t the fuzzy norm N 1. So for each α (0, 1), t>0 such that N 1 (x, t) > 1 α x B t 2 (t + x 1 )(t + x 2 ) > 1 α x = (x 1,x 2 ) B (t + x 1 )(t + x 2 )< t2 1 α t 2 + t( x 1 + x 2 ) + x 1 x 2 < t2 1 α t( x 1 + x 2 )<t 2 α 1 α ( x 1x 2 > 0) x 1 + x 2 < α 1 α t x = (x 1,x 2 ) B α 1 α t x = (x 1,x 2 ) B x 1 <k x = (x 1,x 2 ) B, where k = x 1 < α 1 α t = constant. Thus B is bounded w.r.t 1. Since T is a compact linear operator so T(B)is compact. Since a sequence {x n } is convergent w.r.t N i iff {x n } is convergent w.r.t i (i = 1, 2). So, it follows that T(B)= f T(B).Now consider a sequence {z n } in f T(B)where z n = (y n,w n ). Then it is a sequence in T(B).Then it has a convergent subsequence say {z nk }. Let lim z n k = z (say), where z = (y, w). So, lim { y n k y + w nk w } = 0 k k lim k y n k y =0 and lim k w n k w =0.

9 Fuzzy compact linear operator 223 Now lim N 2(z nk z, t) k = lim N 2((y nk y,w nk w), t) k = lim k (t + y nk y )(t + w nk w ) = t2 t 2 = 1 Thus {z nk } zw.r.t. N 2 and z f T(B).So f T(B)is compact and hence T is a fuzzy compact linear operator. We know that any compact operators on normed linear spaces are bounded [9]. But the following example shows that this is not true for fuzzy compact linear operators on fuzzy normed linear spaces. Example 4.3. Let us consider (l 1, 1 ) and (l 1, 2 ) be two normed linear space where 1 = sup a n and 2 = a n {a n } l 1 and T : (l 1, 1 ) (l 1, 2 ) be n 1 n=1 a compact linear operator which is defined by T {a n }={a n /n} (Example 6.1 [11]). T : (l 1,N 1, 1 ) (l 1,N 2, 2 ) be a mapping defined as above, where N 1 and N 2 are the standard fuzzy norms induced by ordinary norms 1 and 2 respectively, i.e., t,t>0, N i (x, t) = t + x i 0, t 0 for i = 1, 2 and 1 = 2 = min (Example 2.1 [1]), x ={a n }. Let B be a bounded set in l 1 w.r.t the fuzzy norm N 1. So for each α (0, 1) t>0 such that N 1 (x, t) > 1 α x B t > 1 α, x B. t + x 1 t > 1 α, x ={a n } B t + a n 1 sup a n < t n 1 1 α t = tα 1 α, x ={a n} B sup a n <k, x ={a n } B, where k = tα n 1 1 α = (constant) Thus B is bounded w.r.t 1. Since T is a compact linear operator so T(B) is compact. Since a sequence {x n } is convergent w.r.t N i iff {x n } is convergent w.r.t i (i = 1, 2). So, it follows that T(B)= f T(B).Consider a sequence {a n } in f T(B). Then it is a sequence in T(B). Then it has a convergent subsequence say {a nk }. Let lim a n k = a (say) i.e., lim a n k a 2 = 0. k k t 2

10 224 S. Chatterjee, T. Bag, S.K. Samanta Now lim N 2(a nk a,t) k t = lim = t k t + a nk a 2 t = 1. Thus {a n k } has a convergent subsequence w.r.t. N 2 and a f T(B).So f T(B)is compact. Thus T is a fuzzy compact linear operator. If possible let T be fuzzy bounded then for each α (0, 1) M α > 0 such that N 1 (x, t M α ) 1 α N 2 (T x,s) α s >t, t >0, x l 1. Let t>0 and α (0, 1) be fixed. Now Now, N 1 (x, t ) 1 α sup a n t M α n 1 M α s >t,n 2 (T x,s) = α 1 α a n N, n 0 s s + n=1 a n /n α a n /n α n=1 1 α { 1, n= 1,...,k = N>0 = (constant). s = K = (constant) (a) Now let a n =. Then {a n } X. Now from (a) we have, 0, n>k K k N, as k, then K, which is a contradiction. Thus T is not a fuzzy bounded linear operator. k n=1 1/n Theorem 4.4. Let T : (X, N 1, 1 ) (Y, N 2, 2 ) be a linear operator and 2 is continuous at (1, 1). Then T is fuzzy compact linear operator iff it maps every bounded sequence {x n } in (X, N 1, 1 ) onto a sequence {T(x n )} in (Y, N 2, 2 ) which has a convergent subsequence. Proof. Suppose T is fuzzy compact linear operator and {x n } be a bounded sequence in (X, N 1, 1 ). Then the {f T(x n ); n N} is compact. So by definition of compactness, {T(x n )} has a convergent subsequence in (Y, N 2, 2 ). Conversely, let A be a bounded subset of X w.r.t the fuzzy norm N 1. We have to show that f T (A) is compact subset of Y. Let {x n } be a sequence in f T (A). Then by Lemma?? a sequence {z n } in T (A) such that Now let T(y n ) = z n, where y n A. lim N 2(z n x n,t)= 1 t>0 (1) n

11 Fuzzy compact linear operator 225 Now by the given condition {T(y n )} has a convergent subsequence say {T(y nk )}. Let T(y nk ) = z nk z w.r.t fuzzy norm N 2, for some z Y. Thus, lim N 2(z nk z, t) = 1 t>0 (2) k Form (1) and (2), we have N 2 (x nk z, t) N 2 (x nk z nk,t/2) 2 N 2 (z nk z, t/2) lim N 2(x nk z, t) k lim N 2(x nk z nk,t/2) 2 lim N 2(z nk z, t/2) k k = = 1(As 2 is continuous at (1,1)) Thus every sequence in f T (A) has a convergent subsequence. So f T (A) is compact set. Hence, T is a fuzzy compact linear operator. Remark 4.5. The set of all fuzzy compact linear operators from (X, N 1, 1 ) to (Y, N 2, 2 ) is denoted by F(X,Y). Theorem 4.6. Let (X, N 1, 1 ) and (Y, N 2, 2 ) be two fuzzy normed linear spaces and 2 is continuous at (1, 1). Then the set of all fuzzy compact linear operators from X to Y is a linear subspace of L(X, Y ), where L(X, Y ) is the collection of all linear operators from (X, N 1, 1 ) to (Y, N 2, 2 ). Proof. Let T 1,T 2 F(X,Y)and {x n } be a sequence in X which is bounded. Then, the sequence {T 1 (x n )} has a convergent subsequence by Theorem 4.4 say {T 1 (x nk )}. Again by the Theorem??, {T 2 (x nk )} also has a convergent subsequence say {T 2 (z nk )}. Hence {T 1 (z nk )} and {T 2 (z nk )} both are convergent w.r.t the fuzzy norm N 2. Let {T 1 (z nk )} u and {T 2 (z nk )} v, u, v Y. Now for any t>0, lim N 2((T 1 + T 2 )(z nk ) u v, t) k lim N 2(T 1 (z nk ) u, t/2) 2 lim N 2(T 2 (z nk ) v, t/2) k k = = 1. This implies that T 1 + T 2 F(X,Y).Now by the similar argument of Theorem 8 of [6] it can be shown that kt F(X,Y), k R {0}. Theorem 4.7. Let (X, N, ) be a fuzzy normed linear space and is continuous at (1, 1). If T is a fuzzy compact linear operators and S is a fuzzy bounded linear operator from (X, N, ) to (X, N, ), then ST and TSare fuzzy compact linear operators. Proof. Since is continuous at (1, 1) then by Theorem 4.4 T is fuzzy compact linear operator iff it maps every bounded sequence {x n } in (X, N, ) onto a sequence {T(x n )} in (X, N, ) which has a convergent subsequence. Now let {x n } be a bounded sequence

12 226 S. Chatterjee, T. Bag, S.K. Samanta in (X, N, ). Then {T(x n )} has a convergent subsequence in (X, N, ) say {T(x nk )}. Since S is fuzzy bounded so S is fuzzy continuous by Theorem Thus {ST (x nk )}= {S(T (x nk ))} is convergent in (X, N, ) and hence ST is fuzzy compact. Again since S is fuzzy bounded linear operator then for each α (0, 1) M α such that the relation (2.15.1) holds. Let {x n } be a bounded sequence in (X, N, ). Then for each α (0, 1), t>0 such that N(x n,t)>1 α, n N. Let α 0 (0, 1) and β = 1 α 0 (0, 1). Now since S is fuzzy bounded operator then by the relation (2.15.1), M β > 0 such that {t >0 : N(S(x),t) β} M β ( {t >0 : N(x,t) 1 β}) x X holds. Since {x n } is bounded sequence, so for β (0, 1), t > 0 such that N(x n,t )>1 β, n = 1, 2,... {t>0 : N(x n,t) 1 β} <t, n = 1, 2,... Thus by the relation (2.15.1), n = 1, 2,... {t >0 : N(S(x n ), t) β} M β t N(S(x n ), M β t ) β N(S(x n ), t ) 1 α 0, where t = M β t and β = 1 α 0. As α 0 is arbitrary, so {S(x n )} is bounded in (X, N, ). Since T is fuzzy compact linear operator so {TS(x n )}={T(S(x n ))} has a convergent subsequence. TSis fuzzy compact linear operator. Theorem 4.8. Let T : (X, N 1, 1 ) (Y, N 2, 2 ) be a linear operator with 2 be lower semicontinuous and continuous at (1, 1). If T is fuzzy bounded and dim T(X)< then T is a fuzzy compact linear operator. Proof. Let {x n } be a bounded sequence in (X, N 1, 1 ). Since T is fuzzy bounded so for each α (0, 1), M α > 0 such that {t>0 : N 2 (T (x), t) α} M α ( {t >0 : N 1 (x, t) 1 α}) x X (1) Since {x n } is bounded then for each 0 <α<1, t>0such that N 1 (x n,t)>1 α n N. Let α 0 (0, 1). Then β = 1 α 0 (0, 1), and for β (0, 1) t 0 > 0 such that N 1 (x n,t 0 )>1 β n N, n N.

13 Fuzzy compact linear operator 227 {t>0; N 1 (x n,t) 1 β} t 0 M β ( {t >0; N 1 (x n,t) 1 β}) M β t 0 = K>0, n N. (a constant) From (1), {t >0; N 2 (T (x n ), t) β} K, n N. N 2 (T (x n ), K) β = 1 α 0, n N. Since α 0 (0, 1) is arbitrary, thus {T(x n )} is bounded and so is {f T(x n )} by Lemma 3.5. Then {f T(x n )} is compact by Theorem Thus any sequence in {f T(x n )} has a convergent subsequence w.r.t fuzzy norm N 2. So, {T(x n )} has a convergent subsequence w.r.t fuzzy norm N 2. T is a fuzzy compact linear operator. Theorem 4.9. Let T : (X, N 1, 1 ) (Y, N 2, 2 ) be a linear operator with N 1 satisfy (N6) and 1, 2 be both lower semicontinuous and 2 be continuous at (1, 1). If dim X, then T is a fuzzy compact linear operator. Proof. Since dim X< and 1 is lower semicontinuous this implies that T is a fuzzy bounded linear operator by Theorem 3.6. Again dim T(X) <dimx<, by Theorem 4.8. Thus T is a fuzzy compact linear operator. 5. Conclusion In the classical functional analysis, compact linear operators defined on normed spaces, are very important in applications. They play a crucial role in integral equations and in various problems of mathematical physics. Hence, the fuzzy type of them can also play an important part in the fuzzy areas. In this paper, a theory of fuzzy compact linear operators on fuzzy normed linear space are studied. Some basic properties of fuzzy compact linear operators are established. Hence, some classical results on compact linear operators are generalized. References [1] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, The Journal of Fuzzy Mathematics 11(3)(2003) [2] George J. Klir, Bo Yuan, Fuzzy Sets and Fuzzy Logic, Printice-Hall of India Private Limited, New Delhi (1997). [3] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, Annals of Fuzzy Mathematics and Informatics 6(2)(2013) [4] T. Bag, S. K. Samanta, Some observations on completeness and compactness in fuzzy normed linear spaces, Annals of Fuzzy Mathematics and Informatics, The Journal of fuzzy mathematics 24(2)(2016) [5] T. Bag, S. K. Samanta, Operator s fuzzy norm and some properties, Fuzzy information and engineering, 7(2015)

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