Chaos, Solitons and Fractals 34 (2007) 1584–1589 www.elsevier.com/locate/chaos Fuzzy compact linear operators Fatemeh Lael, Kourosh Nourouzi * Department of Mathematics, K.N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran Accepted 25 April 2006 Communicated by Prof. Gerando Iovane Abstract In this paper, we introduce fuzzy compact linear operators between fuzzy normed spaces and investigate some important general properties of them. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction It is well known that in classical mechanics the algebra of observables is the commutative algebra of functions on some space, while in quantum mechanics or quantum field theory, the observables are operators on a Hilbert space, and the algebra of operators on a Hilbert space is non-commutative. On the one hand, this non-commutativity of operators on a Hilbert space provides a precise formulation of the uncertainty principle: there are operator solutions to equations like rs  sr = 1. This equation has no commutative counterpart. In fact, it has no solution in operators r, s acting on a finite dimensional space. Thus in the dynamics of quantum theory, one must work with operators rather than functions and, more precisely, operators on infinite dimensional spaces. Hence, due to the significance of operators, investigating specific operators is quite essential. On the other hand, the usual uncertainty principle of Heisenberg leads to generalized uncertainty principle, which has been motivated by string theory and non-commutative geometry. In strong quantum gravity regime spacetime points are determined in a fuzzy manner. Thus impossibility of determining position of articles gives spacetime a fuzzy structure [5,4,10–12]. Because of this fuzzy structure, position space representation of quantum mechanics breaks down and therefore a generalized Hilbert space of quasi-position eigenfunctions is required [11]. Thus, one needs to discuss on a new family of fuzzy type operators. Since fuzzy mathematics and fuzzy physics along with the classical ones are constantly developing, the phenomena being present in these classical sciences causes us to invent their fuzzy peers. In the classical functional analysis, compact linear operators defined on normed spaces, specially on Hilbert spaces, are very important in applications. They play a crucial role in integral equations and in various problems of mathematical physics, for instance. Hence, the fuzzy type of them can also play an important part in the new fuzzy areas. In this paper, after an introduction to fuzzy normed spaces, we introduce fuzzy compact linear operators between fuzzy normed spaces and investigate their important general properties. * Corresponding author. Fax: +98 21 22853650. E-mail address: nourouzi@kntu.ac.ir (K. Nourouzi). 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.04.055 F. Lael, K. Nourouzi / Chaos, Solitons and Fractals 34 (2007) 1584–1589 1585 2. Preliminaries The notion of fuzzy norm on a linear space was first introduced by Katrasas [9]. Feblin [6] gave an idea of a fuzzy norm on a linear space whose associated metric is Kalva type [7]. Cheng and Menderson [3] considered a fuzzy norm on a linear space whose associated metric is Kramosil and Michalek type [8]. Felbin’s definition of a fuzzy norm of a linear operator between two fuzzy normed spaces was generalized by Xiao and Zhu [13]. Bag and Samanta [1] introduced a notion of boundedness of a linear operator between fuzzy normed spaces, and studied the relation between fuzzy continuity and fuzzy boundedness. They also considered fuzzy bounded linear functionals, the concept of fuzzy dual spaces, and established some fundamental theorems in the area of fuzzy functional analysis. We give below some basic preliminaries required for this paper. We consider fuzzy normed spaces defined by Bag and Samanta [1]. Throughout of this paper X will be a real vector space and Rþ denotes the set of all positive real numbers. Definition 1. A fuzzy subset N of X  R is called a fuzzy norm on X if the following conditions are satisfied for all x, y 2 X, and c 2 R: (N1) N(x, t) = 0 for all non-positive real number t, (N2) N(x, t) = 1 for all t 2 Rþ if and only if x = 0,   (N3) Nðcx; tÞ ¼ N x; jcjt , for all t 2 Rþ and c 5 0, (N4) N(x + y, s + t) P min{N(x, s), N(y, t)}, for all s; t 2 R, (N5) N(x, Æ) is a non-decreasing function on R, and supt2R N ðx; tÞ ¼ 1. The pair (X, N) is said to be a fuzzy normed space. Theorem 1 [1]. Let (X, N) be a fuzzy normed space, and that (N6) N(x, t) > 0, for all t 2 Rþ implies that x = 0. V To each a 2 (0, 1) define kxka ¼ ft 2 Rþ : Nðx; tÞ P ag. Then {k Æ ka : a 2 (0, 1)} is an ascending class of norms on X. These norms are called a-norms on X corresponding to the fuzzy norm N on X. Definition 2. Let (X, N) be a fuzzy normed space, and {xn} be a sequence in X. Then the sequence {xn} is said to be N fuzzy convergent to x 2 X and denoted by xn ! x or xn ! x if limn!1N(xn  x, t) = 1, for all t > 0. Definition 3 [2]. Let T : (X, N1) ! (Y, N2) be a linear operator, where (X, N1) and (Y, N2) are fuzzy normed spaces. 1. The operator T is called weakly fuzzy continuous at z 2 X if for any  > 0, and a 2 (0, 1) there exists d > 0 such that for all x 2 X if N1(x  z, d) P a then N2(T(x)  T(z), ) P a. If T is weakly fuzzy continuous at each point of X, then T is said to be weakly fuzzy continuous on X. 2. The T is called weakly fuzzy bounded on X if for every a 2 (0, 1) there exists ma > 0 such that if  operator  N 1 x; mta P a, then N2(T(x), t) P a, for all x 2 X and t 2 R. We will denote the set of all weakly fuzzy bounded operators from (X, N1) into (Y, N2) by F 0 (X, Y). It is proved that F 0 (X, Y) is a vector space (see [2, Theorem 5.1]). 3. The operator T is called strongly fuzzy continuous at z 2 X if for every  > 0 there exists d > 0 such that N2(T(x)  T(z), ) P N1(x  z, d), for all x 2 X. If T is strongly fuzzy continuous at each point of X, then T is said to be strongly fuzzy continuous on X. 4. The operator T is said
to be strongly fuzzy bounded on X if there exists a positive real number M such that N 2 ðT ðxÞ; tÞ P N 1 x; Mt , for all x 2 X and t 2 R. Theorem 2 [2]. Let (X, N1) and (Y, N2) be two fuzzy normed spaces and T : (X, N1) ! (Y, N2) a linear operator. Then T is weakly (strongly) fuzzy continuous if and only if T is weakly (strongly) fuzzy bounded. The following condition of a fuzzy norm N will be needed: (N7) For any nonzero element x, N(x, Æ) is a continuous function on R and strictly increasing on {t : 0 < N(x, t) < 1}. 1586 F. Lael, K. Nourouzi / Chaos, Solitons and Fractals 34 (2007) 1584–1589 Theorem 3. [2]. Let (X, N1) and (Y, N2) be two fuzzy normed spaces satisfying (N6) and (N7). Then the linear operator T : (X, N1) ! (Y, N2) is weakly fuzzy bounded if and only if T is bounded with respect to a-norms corresponding to N1 and N2. Theorem 4 [2]. Let T : (X, N1) ! (Y, N2) be a linear operator, where (X, N1) and (Y, N2) are fuzzy normed spaces. If T is strongly fuzzy continuous on X, then T is sequentially continuous, i.e., for any sequence {xn} in X with xn ! x implies that T(xn) ! T(x). Definition 4. A subset B in a fuzzy normed space (X, N) is called fuzzy compact if each sequence of elements of B has a fuzzy convergent subsequence. Definition 5 [2]. A subset B of a fuzzy normed space X is said to be fuzzy bounded if there are t > 0 and 0 < r < 1 such that N(x, t) > 1  r, for all x 2 B. 3. Fuzzy compact operators In this section, we introduce fuzzy compact operators between fuzzy normed spaces and examine some fundamental properties of these operators. Definition 6. The fuzzy closure of a subset E of a fuzzy normed space (X, N) is denoted by E and defined by the set of all x 2 X such that there is a sequence {xn} of elements of E with xn ! x. We say that E is fuzzy closed if E ¼ E. Definition 7. Let (X, N1) and (Y, N2) be fuzzy normed spaces. A linear operator T : (X, N1) ! (Y, N2) is called a fuzzy compact operator if for every fuzzy bounded subset M of X the subset T(M) of Y is relatively fuzzy compact, that is the fuzzy closure of T(M) is a fuzzy compact set. Example 1. Let (X, k Æ k1) and (Y, k Æ k2) be two ordinary normed spaces, and T : X ! Y be a compact operator. Then it is easy to see that T : (X, N1) ! (Y, N2) is a fuzzy compact operator, where N1 and N2 are the standard fuzzy norms induced by ordinary norms k Æ k1 and k Æ k2, respectively, i.e., ( t ; t > 0; N i ðx; tÞ ¼ tþkxki 0; t 6 0; for i = 1, 2. Example 2 (Integral operator). Let C[0, 1] be the set of all real valued continuous functions on [0, 1] with the fuzzy norm NðuðxÞ; tÞ ¼ tþsup t juðxÞj, where u 2 C[0, 1] and t > 0. If k(x, y) with x, y 2 [0, 1] is a real valued continuous function, x2½0;1 then the operator T : C[0, 1] ! C[0, 1] defined by Z 1 kðx; yÞuðyÞ dy; ðT uÞðxÞ ¼ 0 where u 2 C[0, 1] is a fuzzy compact operator. Theorem 5. Let T : (X, N1) ! (Y, N2) be a linear operator. Then T is fuzzy compact if and only if it maps every fuzzy bounded sequence {xn} in X onto a sequence {T(xn)} in Y which has a fuzzy convergent subsequence. Proof. Suppose that T is a fuzzy compact operator, and {xn} is a fuzzy bounded sequence in (X, N1). The fuzzy closure of fT ðxn Þ : n 2 Ng is a fuzzy compact set. So {T(xn)} has a fuzzy convergent subsequence by definition. Conversely, let A be a fuzzy bounded subset of (X, N1). We show that the fuzzy closure of T(A) is fuzzy compact. Let {xn} be a sequence in the
fuzzy closure of T(A). For given  > 0, n 2 N and t > 0, there exists {yn} in T(A) such that N 2 xn  y n ; 2t > 1  . Let yn = T(zn), where zn 2 A. Since A is a fuzzy bounded set, so is {zn}. On the other hand, because T is fuzzy compact oper
ator, T(zn) has a fuzzy convergent subsequence fy nk g ¼ fT ðznk Þg. Let y nk ! y for some y 2 Y. Hence N 2 y nk  y; 2t > 1  , for all nk > n0. We have N 2 ðxnk  y; tÞ P minfN 2 ðxnk  y nk ; t=2Þ; N 2 ðy nk  y; t=2Þg > 1  ; for all nk > n0. Now fxnk g is a fuzzy convergent subsequence of {xn}. Thus the fuzzy closure of T(A) is a fuzzy compact set. h Lemma 1. Let (X, N) be a fuzzy normed space satisfying (N6) and {xn} be a sequence in X. Then limn!1N(xn  x, t) = 1 if and only if limn!1kxn  xka = 0, for all a 2 (0, 1). 1587 F. Lael, K. Nourouzi / Chaos, Solitons and Fractals 34 (2007) 1584–1589 N Proof. Suppose that xn ! x. Choose a 2 (0, 1) and t > 0. There exists k 2 N such that N(xn  x, t) > 1  a, for all n P k. It follows that kxn  xk1a 6 t, for all n P k. Thus kxnV xk1a ! 0. Conversely, let kxn  xka ! 0, for every a 2 (0, 1). Fix a 2 (0, 1), and t > 0. There exists k 2 N such that fr > 0 : N ðxn  x; rÞ P 1  ag < t, for all n P k. It implies that N N(xn  x, t) P 1  a, for all n P k, i.e., xn ! x0 . h Definition 8. Let (X, N) be a fuzzy normed space. We define the following subset of X: Ba ½x; r ¼ fy 2 X : Nðx  y; rÞ P ag; where x 2 X, a 2 (0, 1), and r > 0. Theorem 6. Let (X, N) be a fuzzy normed space satisfying (N6) and N(x, Æ) is a continuous function on R. Then X is finite dimensional if and only if Ba[x, r] is a fuzzy compact set in X, for each a 2 (0, 1), and r > 0. Proof. Let Aa[x, r] = {y 2 X : kx  yka 6 r}, where a 2 (0, 1) and r > 0. We first show that Ba[x, r] = Aa[x, r]. If kx  yka 6 r, y 2 Aa[x, r]. Now if y 2 Aa[x, r], then kx  yka 6 r, or y 2 Ba[x, r], then N(x  y, r) P a. Since V V ft > 0 : N ðx  y; tÞ P ag 6 r. If ft > 0 : N ðx  y; tÞ P ag < r, then N(x  y, r) P a. Thus y 2 Ba[x, r]. If V ft > 0 : N ðx  y; tÞ P ag ¼ r, there is {tn} such that tn ! r, and N(x  y, tn) P a. By the continuity of N(x, Æ) we obtain N(x  y, r) = limn!1N(x  y, tn) P a. Hence y 2 Ba[x, r]. Consequently Aa[x, r] = Ba[x, r]. Suppose now that dim X < 1, x 2 X, and r > 0. Choose the sequence {xn} in Ba[x, r]. It is clear that Aa[x, r] is a kka compact subset of (X, k Æ ka). Hence there is a subsequence fxnk g of {xn} and v 2 Aa[x, r] such that xnk ! v. Because in kkb N finite dimensional spaces all norms are equivalent, xnk ! v, for all b 2 (0, 1). Thus by Lemma 1 we obtain xnk ! v. Since Ba[x, r] = Aa[x, r], we have v 2 Ba[x, r]. Conversely, let Ba[x, r] be fuzzy compact. To show that X is finite dimensional, it suffices to prove that Aa[x, r] is compact with respect to a-norm. Choose a sequence {xn} of Aa[x, r]. Since Ba[x, r] is fuzzy compact, it has a fuzzy convergent subsequence fxnk g. Lemma 1 implies that fxnk g is convergent under k Æ ka. Thus Aa[x, r] is compact in normed space (X, k Æ ka). This shows that X is finite dimensional. h Lemma 2. Let T : (X, N1) ! (Y, N2) be a fuzzy compact operator, where (X, N1) and (Y, N2) are fuzzy normed spaces satisfying (N6). Then T : ðX ; k  k1a Þ ! ðY ; k  k2a Þ is an ordinary compact operator for all a 2 (0, 1). Proof. We show that for each bounded sequence {xn} in ðX ; k  k1a Þ, the sequence {T(xn)} has a convergent subsequence in ðY ; k  k2a Þ. Let {xn} be a bounded sequence in ðX ; k  k1a Þ. There exists M > 0 such that kxn k1a < M for all n 2 N. Hence N1(xn, M) P a, for all n, that is {xn} is fuzzy bounded. Thus {T(xn)} has a fuzzy convergent subsequence fT ðxnk Þg. By Lemma 1, fT ðxnk Þg is convergent under k  k2a . h Theorem 7. Let (X, N1) and (Y, N2) be two fuzzy normed spaces satisfying (N6) and (N7). Then (a) Every fuzzy compact linear operator T:(X, N1) ! (Y, N2) is weakly fuzzy continuous. (b) If dim X = 1, then the identity operator I : (X, N1) ! (X, N1) is not a fuzzy compact operator. Proof. (a) Choose a 2 (0, 1). Let k  k1a and k  k2a are a-norms on X and Y corresponding to the fuzzy norms N1 and N2, respectively. By Lemma 2, T : ðX ; k  k1a Þ ! ðY ; k  k2a Þ is a compact operator. Since compact operator is bounded, there exists Ma > 0 such that kT ðxÞk2a 6 M a kxk1a . Hence T is weakly fuzzy bounded by Theorem 3. Now Theorem 2 implies that T is weakly fuzzy continuous.(b) The identity operator I maps Ba[0, 1] to itself. Suppose on the contrary that I is a fuzzy compact operator. Then Ba ½0; 1 is fuzzy compact, for all a 2 (0, 1). Now Ba ½0; 1  Aa ½0; 1 ¼ Ba ½0; 1 implies that Ba[0, 1] is closed and so fuzzy compact. Thus X is finite dimensional by Theorem 6, which is a contradiction. h Theorem 8. Let (X, N1) and (Y, N2) be two fuzzy normed spaces. Then the set of all fuzzy compact linear operators from X into Y is a linear subspace of F 0 (X, Y). Proof. Suppose that T1 and T2 are fuzzy compact linear operators from X into Y. Let {xn} be any fuzzy bounded sequence in X. The sequence {T1(xn)} has a fuzzy convergent subsequence fT 1 ðxnk Þg. The sequence fT 2 ðxnk Þg also has a fuzzy convergent subsequence {T2(zn)}. Hence {T1(zn)} and {T2(zn)} are fuzzy convergent sequences. Let T1(zn) ! u, and T2(zn) ! v. If t > 0, we have n   t t o lim N 2 ðT 1 þ T 2 ðzn Þ  u  v; tÞ P lim min N 2 T 1 ðzn Þ  u; ; N 2 T 2 ðzn Þ  v; : n!1 n!1 2 2 Thus limn!1N2(T1 + T2(zn)  u  v, t) = 1, for all t > 0 . This implies that T1 + T2 is a fuzzy compact operator. Now if   T ðxnk Þ ! y, then limn!1 N 2 ðaT 1 ðxnk Þ  ay; tÞ ¼ limn!1 N 2 T 1 ðxnk Þ  y; jajt ¼ 1, for all a 2 R n f0g, and t > 0. Hence aT1 is also a fuzzy compact operator. h 1588 F. Lael, K. Nourouzi / Chaos, Solitons and Fractals 34 (2007) 1584–1589 Theorem 9. Let (X, N) be a fuzzy normed space, T : (X, N) ! (X, N) be a fuzzy compact linear operator, and S : (X, N) ! (X, N) be a strongly fuzzy continuous linear operator. Then ST and TS are fuzzy compact operators. Proof. Let {xn} be any fuzzy bounded sequence in X. Then {T(xn)} has a fuzzy convergent subsequence fT ðxnk Þg. Let limn!1 T ðxnk Þ ¼ y. Since S is strongly fuzzy continuous, by Theorem 4 we have SðT ðxnk ÞÞ ! SðyÞ. Hence ST(xn) has a fuzzy convergent subsequence. This proves ST is fuzzy compact. Now to show that TS is fuzzy compact, choose any fuzzy bounded sequence {xn}. There exist t0 > 0 and r0 2 (0, 1) such that N1(xn, t0) > 1  r0 for all n P 1. By Theorem 2 we conclude that the operator S is a strongly fuzzy bounded operator. Thus there is M > 0 such that N2(S(xn), t0M) > 1  r0, for all n. It follows that {S(xn)} is fuzzy bounded sequence in S(X). Because T is fuzzy compact, {T(S(xn))} has a fuzzy convergent subsequence. This completes the proof. h Lemma 3. Let (X, N) be a fuzzy normed space satisfying (N6), N(x, Æ) be a continuous function on R and dim X < 1. Then each fuzzy bounded sequence {xn} in (X, N) has a fuzzy convergent subsequence. Proof. Let {xn} be a fuzzy bounded sequence in (X, N). There are t0 > 0 and r0 2 (0, 1) such that N(xn, t0) > 1  r0, for all n 2 N. Hence fxn g 2 B1r0 ½0; t0 . By Theorem 6, B1r0 ½0; t0  is a fuzzy compact set, so {xn} has a fuzzy convergent subsequence. h Theorem 10 [2]. Let (X, N1) and (Y, N2) be two fuzzy normed spaces satisfying (N6) and (N7). If T : (X, N1) ! (Y, N2) is a linear operator where dim X < 1, then T is weakly fuzzy continuous. Theorem 11. Let (X, N1) and (Y, N2) be two fuzzy normed spaces satisfying (N6) and N2(x, Æ) is continuous function on R, and T : (X, N1) ! (Y, N2) a linear operator. (a) If T is weakly fuzzy bounded and dim T(X) < 1, then T is a fuzzy compact operator. (b) In addition if (X, N1) and (Y, N2) satisfying (N7) and dim X < 1, then T is a fuzzy compact operator. Proof. (a) Let {xn} be a fuzzy bounded sequence of (X, N1). There are t0 > 0 and r0 2 (0, 1) such that N1(xn, t0) > (1  r0) for all n 2 N. Since T is weakly fuzzy bounded, there is M 1r0 > 0 such that for all n,   t0 N 1 ðxn ; t0 Þ P 1  r0 ) N 2 T ðxn Þ; P 1  r0 : M 1r0 It follows that {T(xn)} is a fuzzy bounded sequence in T(X). 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