Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

Transcription

1 CAUCHY JURNAL MATEMATIKA MURNI DAN APLIKASI Volume 4 (4) (2017), Pages p-issn: ; e-issn: Almost Surjective Epsilo-Isometry i The Reflexive Baach Spaces Miaur Rohma Departmet of Islamic Studies, Faculty of Tarbiyah, STAI Ma Had Aly Al-Hikam, Malag, Idoesia mimiaira@gmail.com ABSTRACT I this paper, we will discuss some applicatios of almost surjective ε-isometry mappig, oe of them is i Loretz space (L p,q -space). Furthermore, usig some classical theorems of w -topology ad cocept of closed subspace α-complemeted, for every almost surjective ε-isometry mappig f : X Y, where Y is a reflexive Baach space, the there exists a bouded liear operator T : Y X with T α such that for every x X. Tf(x) x 4ε Keywords: almost surjectivity; ε-isometry; Loretz space; reflexive space. INTRODUCTION Suppose X ad Y be real Baach spaces. A mappig U: X Y is called a isometry if U(x) U(y) Y = x y X, for all x, y X. Mazur-Ulam showed that if U is a surjective isometry, the U is affie. I other word, a surjective isometry mappig ca be traslated. This result lead to the followig defiitio. Defiitio 1.1. Let X ad Y be real Baach spaces. A mappig f: X Y is called a εisometry if there exists ε 0 such that f(x) f(y) Y x y X ε, for all x, y X. If ε = 0, the 0-isometry is just a isometry. There are other ames of ε-isometry, some of them are earisometry [13], approximate isometry ([1], [7]), ad perturbed metric preserved [3]. To simplify, the orm fuctio is just writte as without metioig the vector space. Defiitio 1.1 above begs a questio, for every ε-isometry f, does always exist a isometry mappig U ad γ > 0 such that f(x) U(x) γε (1.1) for all x X?. This problem was first ivestigated by Hyers-Ulam i 1945 [7] (therefore, ε-isometry problems are also called as Hyers-Ulam problems) ad they foud that for every surjective εisometry mappig f i the Euclidea spaces, there always exists a surjective isometry mappig U Submitted: 13 March 2017 Reviewed: 26 May 2017 Accepted: 30 May 2017 DOI:

2 Almost Surjective Epsilo-Isometry i The Reflexive Baach Spaces satisfyig (1.1) with γ = 10. Some years later, D. G Bourgi studied ε-isometry mappig i the Lebesgue space with γ = 12. I 1983, Gevirtz delivered γ = 5 which was hold for ay Baach space X ad Y [6]. This result was sharpeed by Omladić ad Šemrl [10]. Theorem 1.2. Let X ad Y be Baach spaces ad f : X Y is a surjective ε-isometry, there exists a liear surjective isometry mappig U: X Y such that f(x) U(x) 2ε, for every x X. There are may applicatios of ε-isometry mappig, such as Dai-Dog who use ε-isometry mappig to determie the smoothess ad rotudity of Baach spaces. Šemrl ad Väisälä proposed a defiitio of almost surjective mappig as follows [13]. Defiitio 1.3. Let f : X Y is a mappig, Y 1 is a closed subspace of Y, ad δ 0. A mappig f is called almost surjective oto Y if for every y Y 1, there exists x X with f(x) y δ ad for every u X, there exists v Y 1 with f(u) v δ. Usig Defiitio 1.3, Šemrl ad Väisälä weakeed the surjective coditio become almost surjective [13] (see also [14]). Theorem 1.4. Let E ad F be Hilbert spaces ad f : E F is a almost surjective ε-isometry with f(0) = 0 that satisfies f(e) sup lim if ty < 1, (1.2) t t y =1 the there exists a liear surjective isometry U: E F such that f(x) U(x) 2ε, for every x E. Šemrl ad Väisälä showed that this result is true for E ad F are Lebesgue spaces [13]. Vestfrid decrease the value 1 become ½ i (1.2) ad is valid for ay Baach space [15]. O the other had, Figiel proved that for ay isometry mappig U : X Y with U(0) = 0, there exists a operator φ spa U(x) X with φ = 1 such that φ U = I, the idetity o X [5]. Usig Figiel s theorem, Cheg et. al. Give the followig lemma [2] (see also Qia [11]). Lemma 1.5. If f : R Y is a surjective ε-isometry with f(0) = 0, the there exists φ Y with φ = 1 such that φ, f(t) t 3ε, for ay t R. Usig Vestfrid s theorem ad Lemma 1.5, Mia et. al. showed that the followig lemma is true [12]. Theorem 1.6. Let X ad Y be real Baach spaces, f : X Y is a ε-isometry with f(0) = 0. If the mappig f satisfies almost surjective coditio, i. e. sup y S Y t f(x) lim if ty < 1 t 2, the for every x X, there exists a liear fuctioal φ Y with φ = x = r such that φ, f(x) x, x 4εr, for every x X. Miaur Rohma 168

3 Almost Surjective Epsilo-Isometry i The Reflexive Baach Spaces RESULTS AND DISCUSSION We will discuss some applicatios of Theorem 1.6. Therefore, this paper will be orgaized as follows. Firstly, we will discuss two applicatios of Theorem 1.6, oe of them is to determie the stability of almost surjective ε-isometry i L p,q -spaces (also called as Loretz spaces). Next, we will discuss the stability of almost surjective ε-isometry i reflexive Baach spaces. The used otatios ad termiology are stadard. w (proouced weak star ) topology of the dual of ormed space X is the smallest topology of X such that for every x X, the liear fuctioal x x x o X is cotiuous. The author is greatly idebted to aoymous referee for a umber of importat suggestios that help the author to simplify the proof. 1. Applicatios of Theorem 1.6 Let X ad Y be Baach spaces ad Y B(X, Y). For bouded set C X, is defied T C = sup{ Tx x C}. If there is c C such that Tc = T C, the T attais its supremum over C. Operator T is called as orm-attaiig operator. Theorem 2.1. Let X ad Y be real Baach spaces, f : X Y is a almost surjective ε-isometry with f(0) = 0 satisfies f(x) sup lim if ty < 1 y S Y t t 2, (2.1) the λ i f(x i ) + 4ε λ i x i for every x 1,, x X ad every λ 1,, λ R, where λ i = 1. Proof. Let x 1,, x X ad X = spa (x 1,, x ). Baach spaces X ad Y that satisfy (2.1) are strictly covex [12]. I additio, X ad Y are w -compact covex, so that for every x S X, there exists a liear fuctioal φ x S Y such that φ x, f(x) x, x 4ε, for every x X. Sice f is a almost surjective ε-isometry, the for every y S Y, there exists x 0 X such that f(x 0 ) = y. Therefore, y f(x) t = f(x 0 ) f(tx) t φ x f(x 0 ) φ x f(tx) x x t 0 x + x x 0 x φ x f(x 0 ) φ x f(x) x x 0 x + x x 0 x sup x, x 0 x x, x 0 x + φ x, f(x 0 ) φ x, f(x) x S X = sup x S X x, x 0 x x, x 0 x + φ x, f(x 0 ) f(x) sup x, x 0 x x, x 0 x φ x, f(x 0 ) f(x) x S X Miaur Rohma 169

4 Almost Surjective Epsilo-Isometry i The Reflexive Baach Spaces Recall that X ad Y are strictly covex, so f(x 0 ) f(x) = λ i f(x i ) ad x 0 x = λ i x i. Sice x S X, i. e. x = 1, the x is orm attaiig fuctioal. Hece, sup x, x λ i x i = x ( λ i x i ) = λ i x i. Therefore, S X or λ i f(x i ) λ i x i x, λ i x i = λ i x i λ i x, = λ i x i x, λ i x i 4ε x i λ i f(x i ) + 4ε φ x, λ i f(x i ) x i φ x, f(x i ) φ x, f(x i ) λ i x i. Next, we will discuss a applicatio of Theorem 1.6 i L p,q -spaces. Let L(Ω) is a algebra of equivalece classes of real valued measured fuctio over (Ω,, μ). The distributio δ f (s) for f L(Ω) is defied as δ f (s) = μ( f > s) L p,q -spaces are collectios of all measured fuctio over Ω such that f p,q = ( q p f (t) q t (q p ) 1 dt) 0 where f (t) = if{s > 0 δ f (s) < t}, t > 0. If p = q, the L p,p (Ω) is just Lebesgue space L p (Ω). The followig theorem shows that there exists a isomorphism operator [8] i L p,q -spaces. Theorem 2.2. Suppose that 1 p,q, p q, p 1, p 2. The there exists o weakly sequece {f k } k=1 i L p,q (0,1) such that for some C > 0 ad for ay subsequece {f k } k=1 of {f k } k=1, the estimate N C 1 N 1/p f k k=1 p,q 1 q CN 1/p < holds. Now, usig Theorem 2.2, the author will show that the followig theorem is true. Theorem 2.3. Let X =L p,q ad Y be Baach space. If f : X Y is a almost surjective ε-isometry with f(0) = 0, the there exists a operator S : Y X with S = 1 such that Sf(x) x 4ε, for every x X. Proof. From Theorem 2.2, the B X is a w -closed covex hull of w -exposed poit i L p,q. Let t (0,1) ad x S X. Theorem 1.6 implies that there exists φ t S Y such that φ t, f(x) x t, x 4ε, for every x X. Miaur Rohma 170

5 Almost Surjective Epsilo-Isometry i The Reflexive Baach Spaces Let S : Y X is defied by S(y) = φ t, f(x) e t = x 0. Clearly S = 1 ad 1 Sf(x) x p,q = ( q p (x 0(t) q x(t) q )t (q p ) 1 dt) 0 1 q sup φ t, f(x) x t, x 4ε. t (0,1) 2. Almost Surjective ε-isometry i The Reflexive Baach Spaces A ormed space is reflexive if the atural mappig π(x) from X ito X which is defied by π(x) : f f(x) where f X, is oto X, or X X. Therefore, every reflexive ormed space is Baach space. I additio, every closed subspace of reflexive Baach space is reflexive [4]. Propositio 3.1 (Meggiso [9], Propositio ). Let X be reflexive ormed space. Every fuctioal x X is orm-attaiig fuctioal. The followig defiitios are classic (for more detail see [4]). Let X be Baach space ad M X. The polar set of M is M = { x X x, x 1, for all x M}. Coversely, M = { x X x, x 1, for all x M }. The aihilator of M is M = { x X x, x = 0, for all x M}, ad preaihilator M is M = { x X: x, x = 0 for all x M }. For a almost surjective ε-isometry mappig f : X Y with f(0) = 0 ad ε > 0, C(f) is a closed covex hull of f(x), E is a aihilator of subspace F Y where F is a set of all bouded fuctioals over C(f), ad M ε = {φ Y : φ is bouded by βε over C(f), for β > 0}. Obviously, C(f) is symmetric, the M ε is liear subspace of Y with M ε = =1 C(f). Therefore, E = {kerφ φ M}. The followig lemma is easy to be proved (see [2]). Lemma 3.2. Let Y be Baach space, the followig statemets are equicalece. (1) C(f) E + F for a bouded B Y; (2) M ε is w -closed; (3) M ε is closed. For every almost surjective ε-isometry f, is defied a liear mappig l : X 2 Y by lx = {φ Y : φ, f(x) x, x βε, for β > 0 ad all x X}. Sice l is a liear mappig, the defiitio of M ε implies ad l0 = {φ Y : φ, f(x) 0, x βε} = {φ Y : φ, f(x) βε} = {φ Y : φ is bouded by βε over C(f)} = M ε lx = l(x + 0) = lx + l0 φ + l0 = φ + M ε. Lemma 3.3. If x y, the lx ly =. Proof. Suppose that lx ly is oempty, the there exists φ lx ly. Assume φ x = {φ Y : φ, f(x) x, x βε, for all x X) ad Miaur Rohma 171

6 Almost Surjective Epsilo-Isometry i The Reflexive Baach Spaces φ y = {φ Y : φ, f(x) y, x βε, for all x X). For x y, there exists φ lx ly such that φ x = φ y. This is impossible because φ is a liear fuctioal ad the proof is complete. Let U B(X, Y). The U B(Y, X ) is called a adjoit operator such that Ux, f = x, U f for all x X. Two results below are classic. Propositio 3.4 (Fabia et. al. [4], Propositio 2.6). Let M is a closed subspace of Baach space X. The (X/Y) is isometric with M ad M is isometric with X /M. Theorem 3.5 (Meggiso [9], Propositio ). Let X ad Y be ormed spaces. If U B(X, Y), the U is w -to-w cotiuous. Coversely, if Q is w -to-w liear cotiuous operator from Y to X, the there exists U B(X, Y) such that U = Q. true. Now, usig Propositio 3.4 ad Theorem 3.5, we will show that the followig theorem is Theorem 3.6. Let X ad Y be Baach spaces, f : X Y is a almost surjective ε-isometry with f(0) = 0, ad M = l 0. The (1) Q: X Y /M, which is defied by Qx = lx + M, is liear isometry. (2) If M is w -closed, the Q is a adjoit operator of U: E X with U = 1. Proof. (1) From Lemma 3.3 ad defiitio of l, it is clear that Q is liear. Lemma 3.2 implies M = l 0 = M. ε For every x X, Qx = if{ φ m : φ lx, m M} = if{ φ m : φ lx, m M ε }. From defiitio of lx ad M ε, we have Qx = if{ φ : φ lx }. Theorem 1.6 gives if{ φ : φ lx } x. (3.1) To show the coversely, from defiitio of l, there exists β > 0 such that φ, f(x) x, x βε, for all x X. For ay δ > 0, we ca choose x 0 S X such that For all N, we get As, φ lim sup x, x 0 > x δ. φ, f(x 0) x, x 0 βε φ, f(x 0) = x, x 0 > x δ. Miaur Rohma 172

7 Almost Surjective Epsilo-Isometry i The Reflexive Baach Spaces Sice δ is arbitrary, the φ x. (3.2) From (3.1) ad (3.2), the φ = x which shows that Q is a isometry. (2) Sice M is w -closed, the M = { M} = { M ε } = E. Therefore, Y /M = Y /E. From defiitios of E ad M ε, Propositio 3.4 implies Y /E = E. We will prove that φ is w -to-w cotiuous over the uit ball B X. Let I is a idex set ad α I. Suppose there exists et (x ) B X which is w -covergig to x X. Usig Theorem 1.6, there exists et (φ α ) Y with φ α = x α = r α 1 such that φ α, f(x) x α, x 4εr α, for all x X. Sice (φ α ) is w -compact, the there exists w -limit poit Y such that φ, f(x) x, x 4εr, for all x X Therefore, Qx α = lx α + M = φ α + M is w -covergece to φ + M = Qx. This result shows that Q: B X E is w -to-w cotiuous. From Theorem 3.5, there exists U: E X such that U = Q. Defiitio of E shows that U is a surjective mappig. Sice U = Q is a liear isometry, the U = 1. Before discussig the stability of a almost surjective ε-isometry i reflexive Baach space, we eed some resuls below. Propositio 3.7 (Fabia et. al. [4], Propositio 3.114). Let Y be reflexive Baach space. If M is a closed subspace of Y, the M is Baach space. Corollary 3.8. If Y be reflexive Baach space, the the mappig Q i Theorem 3.6 is a adjoit operator. Proof. Sice Y is reflexive, from Propositio 3.7, M is w -closed. The proof ca be completed by Theorem 3.6. Defiitio 3.9. Let X be Baach space ad 0 α <. A closed subspace M X is called αcomplemeted i X if there is a closed subspace N X with M N = {0} ad projectio P: X M alog N such that X = M + N ad P α. Theorem Let X ad Y be Baach spaces where Y is reflexive, ad f : X Y is a almost surjective ε-isometry, the there exists a bouded liear operator T : Y X ad T α such that Tf(x) x 4ε, for all x X. Proof. Sice Y is reflexive, from Corollary 3.8 ad Theorem 3.6, there exists a surjective operator U: E X with U = 1 such that Q = U. Sice E is α-complemeted i Y, there is a closed subspace F Y ad E F = {0} such that E + F = Y ad projectio P: Y E alog F satisfies P α. Let T = UP, the Furthermore, for all x X ad y Y. Therefore, T = UP U P = P α. Qx, Py = Qx, y Miaur Rohma 173

8 Almost Surjective Epsilo-Isometry i The Reflexive Baach Spaces Qx, Pf(X) x, x = Qx, f(x) x, x 4ε x, (3.3) for all x X ad x X. Defiitio of adjoit operator gives Substitute (3.4) ito (3.3), the for all x X ad x X. I additio, or Qx, Pf(X) = x, Q Pf(X) = x, UPf(X) = x, Tf(X). (3.4) x, Tf(X) x, x = x, Tf(X) x 4ε x x, Tf(X) x x Tf(X) x 4ε x Tf(X) x 4ε for all x X. CONCLUSION I this paper, the author has show simple applicatios of Theorem 1.6. I additio, almost surjective ε-isometries are remai stable i the reflexive Baach spaces. Ope Problem 1. This paper has show that almost surjective ε-isometries are stable i Loretz spaces whe the coditios i Theorem 2.2 are satisfied. Is it possible to geeralize these restrictios? Ope Problem 2. Accordig to the results of this paper, is it possible to determie the stability of almost surjective ε-isometries i the higher dual of Baach spaces? REFERENCES [1] D. G. Bourgi, "Approximate isometries", Bull. Amer. Math. Soc., vol. 52, o. 8, pp , [2] L. Cheg, Y. Dog, ad W. Zhag, "O stability of o-liear o-surjective ε-isometries of Baach spaces", J. Fuct. Aal., vol. 264, o. 3, pp , [3] L. Cheg ad Y. Zhou, "O perturbed metric-preserved mappig ad their stability characteristic", J. Fuct. Aal., vol. 264, o. 8, pp , [4] M. Fabia, P. Habala, P. Hájek, V. Motesios, ad V. Zizler, Baach Space Theory: The Basis for Liear ad Noliear Aalysis, Spriger, New York, [5] T. Figiel, "O oliear isometric embeddig of ormed liear spaces", Bull. Acad. Polo. Sci. Ser. Sci. Math. Astroom. Phys., vol. 16, pp , [6] J. Gevirtz, "Stability of isometries o Baach spaces", Proc. Amer. Math. Soc., vol. 89, o. 4, pp , [7] D. H. Hyers ad S. M. Ulam, "O approximate isometries", Bull. Amer. Math. Soc., vol. 51, o. 4, pp , [8] A. Kuryakov ad F. Sukochev, "Isomorphic classificatio of L p,q -spaces", J. Fuct. Aal. vol. 269, o. 8, pp , Miaur Rohma 174

9 Almost Surjective Epsilo-Isometry i The Reflexive Baach Spaces [9] R. E. Meggiso, A Itroductio to Baach Space Theory, Graduate Text i Mathematics 183, Spriger, New York, [10] M. Omladič ad P. Šemrl, "O oliear perturbatio of isometries", Math. A. vol. 303, o. 4, pp , [11] S. Qia, "ε-isometries embeddig", Proc. Amer. Math. Soc., vol. 123, o. 6, pp , [12] M. Rohma, R. B. E. Wibowo, ad Marjoo, "Stability of a almost surjective epsiloisometry i the dual of real Baach spaces", Aust. J. Math. Aal. Appl., vol. 13, o. 1, pp. 1-9, [13] P. Šemrl ad J. Väisälä, "Nosurjective earisometries of Baach spaces", J. Fuct. Aal., vol. 198, o. 1, pp , [14] I. A. Vestfid, "Almost surjective ε-isometries of Baach spaces", Colloq. Math., vol. 100, o. 1, pp , [15] I. A. Vestfrid, "Stability of almost surjective ε-isometries of Baach spaces", J. Fuct. Aal., vol. 269, o. 7, pp , Miaur Rohma 175