FUZZY NUMERICAL RANGE HILBERT OPERATORS WITH APPLICATIONS IN BEST APPROXIMATION

ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version) doi:_{http://dx.doi.org/10.12732/ijam.v29i5.3}

FUZZY NUMERICAL RANGE HILBERT OPERATORS WITH APPLICATIONS IN BEST APPROXIMATION

M. Iranmanesh1, F. Soleimany2§ 1,2_{Department of Mathematical Sciences}

Shahrood University of Technology IRAN

Abstract: The main purpose of this paper is to introduce the fuzzy numerical range of operator on fuzzy Hilbert space and to study some its properties. Then by applying this concept, we study a version of the problem of the best approximation in fuzzy Hilbert operator spaces.

AMS Subject Classification: 46S40, 41A50, 03E72

Key Words: fuzzy real number, fuzzy norm, fuzzy bounded linear operator, best approximation

1. Introduction

In studying fuzzy topological vector spaces, Katsaras [11] in 1984, first intro-duced the notion of fuzzy norm on a linear space. Felbin [5] introintro-duced an idea of a fuzzy norm on a linear space by assigning a fuzzy real number to each element of the linear space so that the corresponding metric associated this fuzzy norm is of Kaleva type [10] fuzzy metric. In 1994, Cheng and Mordeson [3] introduced another idea of a fuzzy norm on a linear space in such a manner that the corresponding fuzzy metric of it is of Kramosil and Michelek type [12]. Other papers [6, 7, 13] have appeared to study various properties of these types of fuzzy normed linear spaces.

Received: June 16, 2016 c 2016 Academic Publications

Felbin introduced an idea of fuzzy bounded linear operators over fuzzy normed linear spaces and defined fuzzy norm for such an operator. In this paper we use of the definition of fuzzy inner product and a new norm of on operator introduced by M. Saheli, A. Hasankhani and A. Nazari in [9]. They considered the norm of the operator defined in [1] by T. Bag and S.K. Samanta. We introduce the notions of numerical range of linear operators on fuzzy Hilbert spaces and show some properties them. Veeramani [17], Vaezpour and Karimi [16] introduced the concept of t-best approximations in fuzzy metric spaces and fuzzy normed spaces. In this context we investigate another kind of best ap-proximations in fuzzy bounded linear operator spaces over fuzzy Hilbert spaces.

2. Preliminaries

In this section some definitions and preliminary results are given which will be used in our paper.

Definition 1. A mappingη:R_→[0,1] is called a fuzzy real number with

α-level set [η]α ={t:η(t)≥α}, if it satisfies the following conditions:

(N1) there exists t0∈R such thatη(t0) = 1.

(N₂) for eachα_∈(0,1], there exist real numbersη−_{α ≤}η+_α such that the α-level set [η]α is equal to the closed interval [η−α, ηα+].

The set of all fuzzy real numbers is denoted byF(R). Since eachr _∈R can be considered as the fuzzy real number ˜r_∈F(R) defined by

˜

r(t) =

1 if t=r

0 if t₆=r ,

it follows thatR can be embedded in F(R).

Lemma 2. η_∈F(R)if and only if η:R _→[0,1] satisfies (i) η is normal, i.e. there exists anx_{0 ∈}R such thatη(x₀) = 1

(ii) η is fuzzy convex, i.e. η(αx+ (1₋α)y)_≥min_{η(x), η(y)_}for allx, y_∈R

and for allα _∈(0,1)

(iv) lim

t→−∞η(t) = 0,t→lim+∞η(t) = 0

Definition 3. [10] The arithmetic operations_⊕,_⊖,_×and_\onF(R)_×F(R) are defined by

i) (η_⊕γ)(t) = sup_t=s+hmin_{η(s), γ(h)_}.

ii) (η_×γ)(t) = sup_t=s×hmin{η(s), γ(h)}.

iii) (η_⊖γ)(t) = sup_t=s−hmin_{η(s), γ(h)_}.

v) (η

γ)(t) = sup_t=s h

min_{η(s), γ(h)_}.

In this paper, we denote the symbolη2 to replace η_×η.

Definition 4. [10] Letη_∈F(R). Ifη(t) = 0 for allt <0, thenηis called a positive fuzzy real number. The set of all positive fuzzy real numbers is denoted by F+(R), and real number η−_{α ≥}0 for allη _∈F+(R) and all α_∈(0,1].

Lemma 5. [10] Let η, γ _∈ F(R) and [η]α = [η−_α, η_α+], [γ]α = [γ_α−, γ+_α]. Then,

i) [η_⊕γ]α = [η_α−+γ_α−, η_α++γ_α+];

ii) [η_⊖γ]α = [η_α−₋γ_α+, η_α+₋γ_α−];

iii) [η_⊗γ]α = [η_α−γ_α−, η_α+γ_α+] forη, γ _∈F+(R);

v) [˜_η1]α = [_η1+

α, 1 η−

α], if η

−

α >0.

Lemma 6. [10] Let [aα, bα], α _∈ (0,1], be a given family of non-empty intervals. Suppose

(a) for all,0< α1≤α2,[aα2, bα2]⊆[aα1, bα1];

(b) for any increasing sequence_{αk} in (0, 1] converging toα,

[ lim k→∞a

αk_{, lim} k→∞b

αk_{] = [a}α_{, b}α_];

Then the family [aα_{, b}α_{]represents the α-level sets of a fuzzy real number η in}

F(R). Conversely, if [aα, bα], α_∈(0,1] are α-level sets of a fuzzy real number

F(R), then the conditions (a), (b) and (c ) are satisfied.

Definition 7. [10] The absolute value _|η_|of F(R) is defined by |η_|(t) = max_{η(t), η(₋t)_}.

Definition 8. [9] For a positive fuzzy real numberηdefine√η=γ, where [γ]α = [pηα−,

p

ηα+].

Definition 9. [9] The sequence _{ηn} inF(R) converges to η in F(R), if lim

n→∞|ηn−η|α

+_{= 0.}

Definition 10. [10] Let η, δ _∈ F(R) and [η]α = [η−_α, η_α+],[δ]α = [δ_α−, δ+_α]. Define a partial ordering by η _≤ δ, if and only if η_α− _≤ δ_α−, η_α+ _≤ δ_α+ for all

α_∈(0,1].

Definition 11. A subsetA of F(R) is said to be bounded from above if there exists a fuzzy real numberη, called an upper bound ofA, such that ν_≤η

for everyν _∈A. u is called the least upper bound or supremum of A if u is an upper bound andη _≤ηi for all upper boundsηi. Similarly, a lower bound and the greatest lower bound or infimum are defined. Ais said to be bounded, if it is both bounded from above and below.

Definition 12. [18] LetXbe a vector space overR. Assume the mappings

L;R: [0,1]_×[0,1]_→[0,1] are symmetric and non-decreasing in both arguments, and that L(0; 0) = 0 and R(1; 1) = 1. Let _||._|| :X _→ F+(R). The quadruple (X;_||._||;L;R) is called a fuzzy normed linear space (briefly, FNS) with the fuzzy norm _||._||, if the following conditions are satisfied:

(F1) if x6= 0 then inf0<α≤1kxk−α >0,

(F2) if kxk= ˜0 if and only if x= 0,

(F₃) _krx_k= ˜r_kx_k, forx_∈X,r _∈R,

(F4) for x, y∈X,

(F₄R) _||x+y_||(s+t) _≤ R(_||x_k(s);_||x_k(t)) whenever s _{≥ ||}x_k−₁, t _{≥ ||}x_k−₁

and s+t_{≥ ||}x+y_k−₁.

Lemma 13. [18]Let (X;_||._||;L;R) be an FNS.

(1) If L _≤min, then F4L holds whenever ||x+yk−α ≤ ||xk−α +||xk−α, for all

x, y_∈X and α_∈(0,1].

(2) IfL_≥min, then_||x+y_k−_{α ≤ ||}x_k−_α+_||x_k−_α, for allx, y_∈Xand α_∈(0,1], whenever F₄L holds.

(3) If R_≤max, then F4R holds whenever||x+yk+α ≤ ||xk+α +||xk+α, for all

α_∈(0,1].

(4) If R _≥max, then _||x+y_k−_{α ≤ ||}x_k−_α +_||x_k+_α, for all for all x, y _∈X and

α_∈(0,1], wheneverF₄R holds.

In the sequel we fixL(s, t) = min(s, t) and R(s, t) = max(s, t) for all s, t_∈

[0,1] and we write (X,_k._k) or simplyX whenL and R are as indicated above. The following result is an analogue of the triangle inequality.

Theorem 14. [18]In a fuzzy normed linear space(X;_||._||), the condition

F4 is equivalent to

||x+y_{k ≤ k}x_{k ⊕ k}y_k.

Definition 15. [18] Let (X,_kk) be a FNS. A sequencexn∈X is said to converge to x_∈X( lim

n→∞xn=x), if limn→∞||xn−xk

+

α = 0 for all α∈(0,1].

The functionf :X_−→F(R) is called fuzzy convex if for eachx, y_∈X and each λ_∈(0,1),

f(λx+ (1₋λ)y)_≤λf˜ (x) +(1₋˜λ)f(y).

Definition 16. [2] Let f : X _→ F(R) be a fuzzy mapping. f is upper semi-continuous at a point x0 ∈ X if for anyε > 0, there exists a δ > 0 such that

F(x)_≤F(x₀) + ˜ε,

In [14] Yu-Ru Syaua, Ly-Fie Sugiantob, E. Stanley Leec stated that this concept of upper semicontinuity is essentially same as if bothF(x)+

α andF(x)−α are upper semicontinuous at x0 uniformly for α∈(0,1].

Definition 17. SupposeX is a fuzzy normed space andT :X _→X is an operator. The operator T is called fuzzy bounded if for allx_∈X we have

∃k >0, _kT x_{k ≤}˜k_kx_k.

Definition 18. [9] SupposeT :X_→X is a fuzzy bounded. Define fuzzy operator norm ofT in form follows

[_||T_||]α= [sup β<α

sup

kxk−

β≤1

kT x_k−_β,inf_{η_α+: _kT x_{k ≤}η_kx_k}],

for all α_∈(0,1], i.e. _||T_||α = [||T||−α,||T||+α], where

[_||T_||]−_α = sup β<α

sup

kxk−

β≤1

kT x_k−_β,

||T_||+_α = inf_{η_α+: _kT x_{k ≤}η_kx_k}.

Definition 19. [9] LetXbe a vector space overR. A fuzzy inner product on X is a mapping such that for all vectors x, y, z_∈X and allr _∈R, we have:

i) _hx, x_i= ˜0 if and only ifx= 0,

ii) _hx, y_i=_hy, x_i,

iii) _hx+y, z_i=_hx, z_{i ⊕ h}y, z_i,

v) _hrx, y_i= ˜r_hx, y_i, forx_∈X,r _∈R,

iv) for x₆= 0, inf0<α≤1hx, xi−α >0.

The vector spaceX equipped with a fuzzy inner product is called a fuzzy inner product space.

Lemma 20. [9] A fuzzy inner product space X together with its corre-sponding norm_k._k satisfy the Schwarz inequality

3. Main results

Let (H,_hi) be a fuzzy Hilbert space with by inner product _hi. The set of all fuzzy bounded linear operators T :H _→ H is denoted by B(H). At first, we introduce the notions of numerical range of linear operators on fuzzy Hilbert space H. Then we give results of best approximation in the space.

Lemma 21. Let f, g_∈B(H)and _{x_{n} ∈}H be such that_kx_nk−_{α ≤}1, for all α_∈(0,1],

lim

n→∞hf(xn), g(xn)i −

α and _nlim

→∞

hf(xn), g(xn)_i+α kxnk−

2

α

exist. Thenηxn_{, where}

[ηxn_{]α := [sup} β<α

lim

n→∞hf(xn), g(xn)i −

β,_nlim_→∞

hf(xn), g(xn)_i+α kxnk−

2

α

] (1)

is a fuzzy real number.

Proof. Since the inner product_hiis a fuzzy real number and 1

kxnk−α ≥1, for

α_∈(0,1] andβ _∈(0, α) we have

hf(xn), g(xn)_iβ− _{≤ h}f(xn), g(xn)_iβ+_{≤ h}f(xn), g(xn)_iα+_≤ hf(xn), g(xn)i + α kx_nk−α2

.

Now by taking lim and sup onβ, we have

sup β<α

lim

n→∞hf(xn), g(xn)i −

β ≤_nlim_→∞

hf(xn), g(xn)_i+α kxnk−

2

α

,

and thus

ηxn

α −≤ηαxn+. Hence [ηxn_{]α= [η}xn

α −, ηαxn+] is a nonempty interval, for all α∈(0,1]. (a) Let 0< α1 ≤α2. By Lemma 6 (a) for fuzzy real numberhiwe have

sup β<α1

lim

n→∞hf(xn), g(xn)i −

β ≤ sup β<α2

lim

n→∞hf(xn), g(xn)i −

β.

Also since

kxnk−

2

α1 ≤ kxnk

−2

α2 and _nlim_→∞hf(xn), g(xn)i

+

α2 ≤_nlim_→∞hf(xn), g(xn)i

+ α1,

then

lim n→∞

hf(xn), g(xn)_i+α2

kxnk−

2

α2

≤ lim n→∞

hf(xn), g(xn)_i+α1

kxnk−

2

α1

Then

[ηxn α2

−_{, η}xn α2

+_] ⊆[ηxn

α1

−_{, η}xn α1

+_].

(b) Let _{a_k} be a increasing sequence in (0,1] converging to α. Then we have

lim k→∞η

xn αk +

= lim k→∞nlim→∞

hf(xn), g(xn)i+α kxnk−

2

αk

= lim k→∞

h lim

n→∞f(xn),nlim→∞g(xn)i

+ αk k lim

n→∞xnk −2

αk

= h lim

n→∞f(xn),nlim→∞g(xn)i

+ α k lim

n→∞xnk −2

α

= lim n→∞

hf(xn), g(xn)i+α kxnk−

2

α

.

Hence lim k→∞η

xn αk

+ _{= η}xn

α +. Similarly, we can show that lim k→∞η

xn αk

− _{= η}xn α −. Hence,

[ lim k→∞η

xn αk

−_{, lim}

k→∞η

xn αk

+_{] = [η}xn

α −, ηαxn+].

(c) By assumtion, for allα_∈(0,1],

lim

n→∞hf(xn), g(xn)i −

α and lim_n→∞h

f(xn), g(xn)_i+_α kx_nk−α2

exist. Therefore,

−∞< ηxn

α −≤ηαxn+<+∞, for α∈(0,1]. Thus by Lemma 6, ηxn _{is a fuzzy real number.}

SupposeT _∈B(H). We denote by ZT the following set:

ZT :={{xn} ∈H:kxnk=b1 , sup β<α

lim

n→∞kT xnk −

β =||T||−α}. (2)

Definition 22. The numerical range of f relative to g which is denoted by W(f g) is defined as follows:

W(f g) :=_{ηxn

Note thatW(f g) is non-empty, to see that by the definition of the operator norm that there exists a sequence of unit vectors _{xn} ∈H such that {xn} ∈

Zf. Therefore, since lim

n→∞hf(xn), g(xn)i −

α and lim_n→∞

hf(xn),g(xn)i+α

kxnk−

2

α

is a bounded sequence, there exists subsequencesnkα that converge to someλα, βα ∈R. Let

[ηxnkα_{]α := [sup} β<α

lim

n_kα→∞hf(xnkα), g(xnkα)i −

β,_nlim kα→∞

hf(xn_kα), g(xn_kα)i+α kxn_kαk−

2

α

].

By Lemma 21 ηxnkα _{∈W(f g), and thusW(f g) is non-empty.}

Proposition 23. Let f, g_∈B(H). Then W(f g) has an upper bound. Proof. Letηxn _{∈W(f g), forα∈(0,1] andβ < α, from (1) and by Schwarz} inequality we have

hf(xn), g(xn)_i−_{β ≤ |h}f(xn), g(xn)_i|−_{β ≤ ||h}f(xn)_||β−_||g(xn)_||−_β.

By taking limits onn and sup onβ < α we have

ηxn

α −≤(kfk ⊗ kgk)−α.

Similarly we can show that ηxn

α + ≤ (kfk ⊗ kgk)α+. Thus ηxn ≤ kfk ⊗ kgk for

ηxn _{∈W(f g), sokfk ⊗ kgk is a upper bounded for W(f g).}

J. Fang, H. Huang, and Talo improved the expressions of the supremum and infimum, limit inferior and limit superior of a bounded sequence of fuzzy real numbers on the concept Goetschel and Voxman [8]. In the following we give the expressions for endpoints of level sets of the supremum and infimum same as that given in [4, 15].

Definition 24. Let A be a non-empty subset of F(R). If A has a lower bound, then we define its infimum inf(A) by

[inf(A)]α = [sup β<α

inf η∈Aη

−

β,_ηinf_∈Aη+α]

for eachα_∈(0,1]. Dually, ifAhas an upper bound, then its supremum sup(A) is defined by

[sup(A)]α = [sup η∈A

η−_α,inf β<αsup_η∈Aη

+ β].

Definition 25. Let _{un} be a bounded sequence of fuzzy real numbers. Then, its lim sup

n→∞

un=µhas the following expressions:

[µ]α= [lim sup n→∞

u−_n

β,_β<αinf lim sup n→∞

u+_n

β],

for each α_∈(0,1].

Letf, g _∈B(H), andtn∈R such thattm →0+ consider the norm deriva-tives

τ2(f, g) := lim sup m→∞

kf +tmgk2− kfk2 2 ˜tm

.

Lemma 26. Let f, g_∈B(H). Then,

τ2(f, g)+α ≥supW(f g)+α. (4)

Proof. Suppose_{x_{n} ∈}Zf, we get

kf +tmgk+

2

β ≥_nlim_→∞kf(xn) +tmg(xn)k +2

β = lim

n→∞(kf(xn)k

+2

α +t2kg(xn)k+

2

β + 2tmhf(xn), g(xn)i+β) =_kf_k+_β2 +t2_m lim

n→∞kg(xn)k

+2

β + 2tm lim

n→∞hf(xn), g(xn)i

+ β,

hence

kf+tmgk+

2

β − kfk+

2

β 2tm

=tm lim

n→∞kg(xn)k

+2

β + lim_n→∞hf(xn), g(xn)i+β,

setting m_{→ ∞}, and taking lim sup, then

lim sup m→∞

kf+tmgk+

2

β − kfk +2

β

2tm ≥

lim

n→∞hf(xn), g(xn)i

+ β =η

xn β

+_,

for_{xn} ∈Zf.

Therefore

lim sup m→∞

kf +tmgk+

2

β − kfk+

2

β

2tm ≥

sup

{xn}∈Zf

ηxn β

+_.

So

inf

β<αlim supm→∞

kf +tmgk+

2

β − kfk +2

β

2tm ≥

inf β<α_{xsup

n}∈Zf

ηxn β

Since _−kf_k−_β >_−kf_k+_β, then

τ2(f, g)+α ≥supW(f g)+α. (5)

Theorem 27. LetU be a closed convex subset of B(H), f _∈B(H)_\U,

g0∈U and Zf−g0 ={{zn},{−zn}}. If for each h∈U,

supW((f ₋g0)(g0−h))−α ≥0, forα∈(0,1]. (6)

Then _kf₋g0k−α = infh∈Ukf −hk−α.

Proof. Suppose the inequality (6) holds and let _{xn} = {zn} or {−zn}. Then for α_∈(0,1] we have

sup β<α

lim

n→∞h(f−g0)(xn),(g0−h)(xn)i −

β ≥0. (7)

The (7) implies that lim

n→∞h(f −g0)(xn),(g0 −h)(xn)i −

β ≥ 0 because if there existsβ0 such that lim

n→∞h(f−g0)(xn),(g0−h)(xn)i −

β0 <0. Then forλ∈(0, β0]

we have

lim

n→∞h(f −g0)(xn),(g0−h)(xn)i −

λ ≤_nlim_→∞h(f −g0)(xn),(g0−h)(xn)i−β0 <0.

Therefore

sup λ<β0

lim

n→∞h(f −g0)(xn),(g0−h)(xn)i −

β <0. (8)

But it is a contradiction with (7) is true forα_∈(0,1]. Therefore we have

kf₋h_k−_α2 _≥ sup β<α

lim nk→∞k

f(xnk)−h(xnk)k

−2

β

= sup β<α

lim nk→∞k

f(xnk)−g0(xnk) +g0(xnk)−h(xnk)k

−2

β

= sup β<α

[ lim nk→∞k

f(xnk)−g0(xnk)k

−2

β +||g0(xnk)−h(xnk)k

−2

β ]

+ 2_h(f ₋g0)(xnk), g0(xnk)−h(xnk)i

−

β) ≥ sup

β<α lim nk→∞k

f(xnk)−g0(xnk)k

−2

β =kf −g0k

−2

α .

Theorem 28. Let U be a closed convex subset of B(H), f _∈ B(H)_\U

andg0∈U. Ifkf−g0k−α = infh∈Ukf−hk−α forα∈(0,1]. Then for eachh∈U,

τ2(f₋h, h₋g0)−α ≤0.

Proof. i_→ii. Let k, l_∈B(H), then from Lemma 5 we have

(_kb+l 2 k

2_≤[kb 2k ⊕ k

l

2k]

2 _≤ kbk2⊕ klk2 ˜

2 .

With the help of the induction method we have

(_kb1+...+b2k 2k k

2

≤ kb1k

2_{⊕...⊕ kb} 2kk2 ˜

2k ,

now let b1=...=bn=b and b2k_−n=...=b₂k =l, then

(knb+ (2 k_−n)l

2k k

2 _≤ n˜kbk2 ˜ 2k ⊕

˜

(2k_−n)klk2 ˜ 2k .

Since A = _{2nk :k ∈ K} is dense on [0,1], so for λ∈ (0,1] we have ˜ n ˜ 2k →

˜

λ, therefore

kλb+ (1₋λ)l_k2 _≤˜λ_kb_k2_⊕(1₋˜λ)_kl_k2. (9)

Then _k._k2 is a fuzzy convex function. The function φ defined by φ(t) = ktg_˜tk2 is non-decreasing. Because if we put l = 0 from (9), for s, t _∈ R such that 0< s_≤t, we have

(_ksb_k2 _≤˜s tktbk

2_⊕(t−˜ s)

t k0k

2_.

Since _k0_k = ˜0, then φ(s) _≤ φ(t). Now apply this argument to show that the functionφ(t) = kf+tgk_˜t2−kfk2 is non-decreasing. By the assumption fort= 1 we have

kf₋h+t(h₋g0)k−

2

α − kf−hk−

2

α ≤0. Since _kf₋h_k+_{α ≥ k}f₋h_k−_α we get

kf₋h+t(h₋g0)k−

2

α − kf−hk+

2

α ≤0.

So dividing by tm,m _{→ ∞}, and taking lim sup we obtain

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