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Extremal Properties of the Set of Vector-Valued Banach Limits

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Extremal Properties of the Set of Vector-Valued Banach Limits Open Math. 2015; 13: 757–767 Open Mathematics Open Access Research Article Francisco Javier García-Pacheco* Extremal properties of the set of vector-valued Banach limits DOI 10.1515/math-2015-0067 Received June 23, 2015; accepted October 16, 2015. Abstract: In this manuscript we find another class of real Banach spaces which admit vector-valued Banach limits different from the classes found in [6, 7]. We also characterize the separating subsets of ` .X/. For this we first 1 need to study when the space of almost convergent sequences is closed in the space of bounded sequences, which turns out to happen only when the underlying space is complete. Finally, a study on the extremal structure of the set of vector-valued Banach limits is conducted when the underlying normed space is a Hilbert space. We also reach the conclusion that the set of vector-valued Banach limits is not a convex component of BCL.` .X/;X/, provided that X 1 is a 1-injective Banach space satisfying that the underlying compact Hausdorff topological space has isolated points. Keywords: Banach limit, Almost convergence, Group of Isometries, Extremal Structure MSC: 40J05, 46B15, 46B25 1 Introduction In his 1932 book Théorie des opérations linéaires (see [1, p. 34]) Banach extended in a natural way the limit function defined on the space of all convergent real sequences to the space of all bounded real sequences. Definition 1.1 (Banach, [1]). A linear function ' ` R is called a Banach limit exactly when: W 1 ! 1. ' ..x / / 0 if .x / ` and x 0 for all n , n n N n n N n N 2 2 2 1 2 2. ' ..xn/n / ' .xn 1/n if .xn/n ` , and N D C N N 2 1 3. ' 1 12, where 1 denotes the2 constant2 sequence of general term 1. D Banach noticed that a linear function ' ` R is a Banach limit if and only if ' is invariant under the shift W 1 ! operator on ` and lim inf x ' ..x / / lim sup x for all .x / ` . As a consequence, n n n n N n n n n N 1 !1 Ä 2 Ä !1 2 2 1 every Banach limit ' ` R verifies that ' c lim and ' 1. Therefore, Banach limits are norm-1 Hahn- W 1 ! j D k k D Banach extensions of the limit function on c to ` . For a wider perspective on Banach limits we refer the reader 1 to the very interesting contributions given in [2–5]. In [6] the authors fully extend the concept of Banach limit to vector-valued sequences just by noticing that a function ' ` is a Banach limit if and only if ' S , ' is R ` W 1 ! 2 1 invariant under the shift operator on ` , and ' c lim. 1 j D Definition 1.2 (Armario, García-Pacheco, Pérez-Fernández, [6]). Let X be a real normed space. The set of vector- valued Banach limits on X is defined as BL .X/ SN LX . WD X \ If X R, then we simply write BL. The notation used in the definition right above follows. Given a real normed D space X, we have the following: *Corresponding Author: Francisco Javier García-Pacheco: Department of Mathematics, University of Cadiz, Puerto Real 11510, Spain, E-mail: [email protected] © 2015 García-Pacheco, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. 758 F.J. García-Pacheco – CL .` .X/ ;X/ is meant to be the space of all continuous and linear operators from ` .X/ to X when ` .X/ 1 1 1 is endowed with the sup norm. – NX will denote the set of all continuous and linear operators from ` .X/ to X which are invariant under the 1 shift operator on ` .X/. In other words, 1 ˚ NX T CL .` .X/ ;X/ T ..xn/ / T .xn 1/ n N n N WD 2 1 W 2 D C 2 for all .x / ` .X/ : n n N 2 2 1 g It is clear that NX is a closed vector subspace of CL .` .X/ ;X/ which is also closed when CL .` .X/ ;X/ 1 1 is endowed with the pointwise convergence topology. – LX will stand for the set of all continuous and linear operators from ` .X/ to X which are extensions of the 1 limit function on c .X/. In other words, LX T CL .` .X/ ;X/ T c.X/ lim : WD f 2 1 W j D g It is clear that LX is a an affine subset of CL .` .X/ ;X/ (that is, if T; S LX and t R, then tT .1 t/ S 1 2 2 C 2 LX ), which is also closed when CL .` .X/ ;X/ is endowed with the pointwise convergence topology, and 1 every T LX verifies that 2 T T c.X/ lim 1: k k k j k D k k D We will finalize this introduction with the following proposition which will be very useful throughout this manuscript. We recall the reader that if X is a real normed space and x X, then x denotes the constant sequence of 2 general term equal to x. As expected, X denotes the closed subspace of ` .X/ composed of all constant sequences 1 x. Proposition 1.3 (Armario, García-Pacheco, Pérez-Fernández, [7]). Let X be a real normed space. Then: 1. NX T CL .` .X/ ;X/ bps .X/ ker .T / , where bps .X/ stands for the set of sequences in X with D f 2 1 W Â g bounded partial sums. As a consequence, if T NX , then c0 .X/ ker .T /. 2 Â 2. LX T CL .` .X/ ;X/ c0 .X/ ker .T / and T .x/ x for all x X : D f 2 1 W Â D 2 g 3. NX LX T CL .` .X/ ;X/ bps .X/ ker .T / and T .x/ x for all x \ D f 2 1 W Â D X : 2 g 4. BL .X/ ' SN ' .x/ x for all x X : D f 2 X W D 2 g 5. BL .X/ BNX LX and thus it is convex and closed when CL .` .X/ ;X/ is endowed with the pointwise D \ 1 convergence topology. 2 More on the existence of vector-valued Banach limits In [6] it was shown that a large class of real Banach spaces admit vector-valued Banach limits. On the contrary, c0 is free of vector-valued Banach limits. Theorem 2.1 (Armario, García-Pacheco, Pérez-Fernández, [6]). 1. If X a real Banach space 1-complemented in X , then BL .X/ ¿. ¤ 2. BL .c0/ ¿. D In [7] Theorem 2.1 was extended to the following result. Corollary 2.2 (Armario, García-Pacheco, Pérez-Fernández, [7]). Let X be a real Banach space with a monotone Schauder basis. The following conditions are equivalent: – BL .X/ ¿. ¤ – X is 1-complemented in its bidual. Another spaces admitting vector-valued Banach limits are described in the following result. Extremal properties of the set of vector-valued Banach limits 759 Theorem 2.3 (Armario, García-Pacheco, Pérez-Fernández, [7]). Let .X / be a sequence of real normed spaces i i N 2 with BL .Xi / ¿ for every i N. Assume that 'i BL .Xi / for every i N. The map ¤ 2 2 2 P P ' ` Xi Xi i N i N W 1 n2 1 ! 2 n 1 x 'i x i i N n N i n N i N 2 2 7! 2 2 P verifies that ' BL Xi . i N 2 2 1 In this section we find a different class of real Banach spaces which also admit vector-valued Banach limits. Lemma 2.4. Let X be a real normed space. If x SX and x SX are so that x .x/ 1, then x .UX .x; "// 2 2 D .1 "; / for all 0 < " 1. C1 Ä Proof. Because of the convexity of x .UX .x; "// it suffices to show that no element y UX .x; "/ verifies that 2 x .y/ 1 ". Indeed, assume to the contrary that such element y exists. Then D ˇ ˇ ˇ ˇ " ˇx .x/ x .y/ˇ ˇx .x y/ˇ x y < "; D D Ä k k which is a contradiction. Lemma 2.5. Let X be a real normed space. Then: 1. If x SX , then d .x; bps .X// 1. 2 D 2. bps .X/ X 0 , where bps .X/ denotes the closure of bps .X/ in ` .X/. \ D f g 1 Proof. 1. In the first place, we will show that d .x; bps .X// 1. Let x SX such that x .x/ 1. Fix 2 D an arbitrary 0 < " < 1. By applying Lemma 2.4 we have that x .UX .x; "// .1 "; /. Now consider C1 .xn/ bps .X/ such that n N 2 2 .x / x ": n n N k 2 k1 Ä Notice that .x / U .x; "/ therefore x .x / > 1 " for all n . Let M > 0 be such that n n N X n N 2 2 m X xn M Ä n 1 D for all m N. Finally notice that 2 m ! m X X m .1 "/ < x xn xn M Ä Ä n 1 n 1 D D for all m N which is impossible. As a consequence, d .x; bps .X// 1. In order to see that d .x; bps .X// 1 it 2 D only suffices to realize that .x; 0; 0; : : : ; 0; : : : / bps .X/ 2 and .x; 0; 0; : : : ; 0; : : : / x 1: k k1 D 2. It is a direct consequence of 1.
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