Remarks on Orthogonal Convexity of Banach Spaces

It is proved that orthogonal convexity defined by A. JimenezMelado and E. Llorens-F'uster implies the weak Banach-Saks property. Relations between orthogonal convexity and another geometric properties, such as nearly uniform smoothness and property ( P ) , are studied. Introduction. Orthogonal convexity has been introduced by A. Jimenez-Melado and E. Llorens-F'uster (see [3] and [4]) as a geometric property of Banach spaces which implies the fixed point property for nonexpansive mappings. They have shown that various kinds of Banach spaces, such as co, uniformly convex spaces, spaces with the Schur property, the James space, among others have this property. Let E be a Banach space. If A is a nonempty bounded subset of E, we put (A1 = sup{l(zl) : z E A). Moreover for X > 0 and x, y E E let Having a bounded sequence (x,) in E, we use the following notation: 604 D. KUTZAROVA, S. PRUS AND B. SIMS The Banach space E is called orthogonally convex if for each sequence (x ,) in E weakly convergent to zero, with D[(x , ) ] > 0, there exists X > 0 such that Ax [ ( xn >I < D [ ( x n ) I In this paper we prove that orthogonal convexity implies the weak Banach-Saks property. Our second result concerns orthogonal convexity of some direct sums. The remaining part of the paper is devoted to the study of relations between orthogonal convexity and some infinite dimensional geometric properties. Results. Let us recall that a Banach space E has the weak Banach-Saks property (also called the Banach-Saks-Rosenthal property) if every weakly null sequence (2,) in E contains a subsequence (x',) such that the Cesaro means n-' Ci=, xk form a norm-convergent sequence. Theorem 1. Let E be an orthogonally convex Banach space. Then E has the weak Banach-Saks property. Proof. Assume the contrary, that is, there is a weakly null sequence (x ,) in E such that no subsequence has norm-converging Cesaro means. Using a method described in [ I ] one can find a constant c such that for a fixed integer m there exists a sequence (y;),?, with the following properties. 1. There is a sequence of scalars (a;) and an increasing sequence of integers (k,) such that m 5 k l , m kn+l k + 1 ail < c Yn Ci=k,+l and Ci=kn+l I for all n. 2. If n > 1, then for any scalars a, /3. Consider now a sequence (z,) obtained by a rearrangement of the set {yF)m,n2 , . Since the sequence (x,) is weakly null, from 1 it follows that also (z,) converges weakly to zero. By 2 we have 1\z,11 5 1 for all n. Conseqently D[(z,)] 5 2. On the other hand REMARKS ON ORTHOGONAL CONVEXITY OF BANACH SPACES 605 Hence D[(zk)] = 2 and from orthogonal convexity it follows that for some A > 0. We take mo so that & 5 A. Condition 2 shows that for every k I I Y ~ I I 5 1 5 + A ) 5 + A)lIYF Y,"Il 7 whenever m 2 mo and n > 1. In particular this means that y r + y," E MA ( y r , y r ). Therefore applying 2 again, we obtain mo+k limsup llyl + Y ~ O + ~ I I ) t 2 This contradicts (1). In [3] some statements on direct sums are given. Here we present another one. Let (Xi) be a sequence of spaces and Y be a space with a l-unconditional basis (en) (i.e. (en) is an unconditionally monotone basis in the terminology of [2]). Y{(Xi)) denotes the space of all sequences (x(i)), where x(i) E Xi for each i , such that the series C llx(i)llei converges in Y. The norm in Y{(Xi)) is given by the formula Let us also recall that the basis (en) is shrinking if the coefficient functionals e: form a basis of the dual space Y* (see [2] p. 64). Theorem 2. Let (Xi) be a sequence of Banach spaces with the Schur property and Y be an orthogonally convex space with a l-unconditional shrinking basis (ei) . Then the space X = Y {(Xi)) is orthogonally convex. Proof. For any x = (x(i)) E X put 3 = Czl llx(i)J(ei E Y. By the definition llxllx = Ilji.ll y. Let (xn) be an arbitrary weakly null sequence in X with D[(xn)] > 0. Since all Xi have the Schur property, one can see that lim llxm(i)ll = O m-tm 606 D. KUTZAROVA, S. PRUS AND B. SIMS for each i. But the basis of Y is shrinking. Therefore (2) implies that (2,) converges weakly to zero in Y. Fix an integer n. From (2) it is easy to see that Consequently D[(z,)] = D[(x,)] > 0. Since Y is orthogonally convex, there exists X > 0 such that Ax[(?,)] < D[(z,)] . For a fixed n consider an arbitrary z E M4 (x,, x,). Clearly m a { l \ ~ ?rnlI,IIz?nII} 5 (1 + *) IIxm xnII. If x, # 0, then it follows from (2) and (3) that for m large enough. Obviously this inequality holds also in case x, = 0. Therefore for large m we have 2 E Mx(f,, 2,). Hence which means that X is orthogonally convex. Now we turn to the study of relations between orthogonal convexity and another geometric properties. Let us recall a definition introduced in [71. A Banach space X is nearly unifornly smooth (NUS in short) provided for every E > 0 there exists 7 > 0 such that if 0 < t < 7 and (x,) is a basic sequence in the unit ball Bx of X , then there is k > 1 for which We shall say that X is weakly NUS (WNUS in short) if it satisfies the condition obtained from the above definition by replacing "for every E > 0" by "for some E in (0,l)". In [7] it was proved that uniform smoothness implies NUS which in turn implies reflexivity. A slight modification of the proof of Proposition 2.3 [7] shows that WNUS spaces are reflexive. REMARKS ON ORTHOGONAL CONVEXITY OF BANACH SPACES 607 Proposition 3. A Banach space X is WNUS if and only if there exists a constant c E (0 , l ) such that for every basic sequence (2,) in Bx there is k > 1 for which 11x1 + xkll 5 2 c . Proof. Assume that X is a WNUS space. From the definition we obtain some E > 0 and q > 0. Now if t = min{q, 1) and (2,) is a basic sequence in Bx, then there exists k > 1 such that Assume in turn that there exists c E (0 , l ) such that if (2,) is a basic sequence in Bx, then llxl + xk 11 5 2 c for some k > 1. For t E (0 , l ) we have which shows that X is WNUS We also need the next definition. A Banach space X has the weak Opial property provided for every weakly null sequence (x,) in X and every x E X liminf llxn(l 5 liminf IIx + xnII . n + w n + w Theorem 4. If X is a WNUS Banach space with the weak Opial property, then X is orthogonally convex. Proof. Let X be a WNUS space with the weak Opial property. From Proposition 3 we obtain a constant c E (0,l) . Take a positive X < &. Let now (2,) be a weakly null sequence in X with D[(xn)] > 0. For a fixed n there exists a sequence of integers (mk) such that limk+w IM~(xn,xrnk)( limsupm+, IM~(xn,xrn)I . Denote this limit by a,. There is a sequence (zk) with zk E Mx(xn, xrnk), for all k, and limk+, llzkll = a,. Since X is reflexive, we can assume that (zk) converges weakly to some z. 608 D. KUTZAROVA, S. PRUS AND B. SIMS Let d, = limsupk_,, llx, xmk 11. We shall show that where B = (1 $)(I + A) < 1. If (zk) converges to z in norm, then a, = llzll < liminfk,, llzk xmk 1 1 < f (1 + A)dn I Bd, . Let us now consider the case when (zk) does not converge in norm. Passing to a subsequence, we can assume that the elements z, zl z, z2 z, .. . form a basic sequence (see [2] p. 107). Proposition 3 gives us a further subsequence, which we still denote by (zk), such that (5) a,= k-oo lim ~~z+(zk-z)~(~(2-c)max{llzll,liminfllz~-z~~ k r m But 1 1 ~ 1 1 < liminfk,, ((zk xmk(I ; ( I+ A)dn Moreover by the weak Opial property Therefore (5) shows that a, < ;(2 c)(l + A)d, = Bd,. Having established (4), we conclude that Ax[(xk)] = limsup,,, a, < B limsup,,, d, < BD[(xk)] . Let us now mention another geometric properties. In [8] the notion of the uniform Opial property was introduced. A Banach space X has the uniform Opial property provided for every c > 0 there exists r > 0 such that if llxll > c and (x,) is a weakly null sequence in X with llxnll > 1 for all n, then It was proved that if X is a reflexive space with the uniform Opial property, then X* has the fixed point property for nonexpansive mappings. But Lemma 2.2 [8] actually shows that the space X* is WNUS and has the weak Opial property. Therefore we obtain the following corollary of Theorem 4. REMARKS ON ORTHOGONAL CONVEXITY OF BANACH SPACES 609 Corollary 5. If X is a red exive space with the uniform Opial property, then X * is orthogonally convex. The next property related to WNUS was defined in [9] as a generalization of uniform convexity. It was called property (P), but instead of quoting its definition we prefer to recall the following result (see [5]). A Banach space X has property (P) if and only if for every E > 0 there exists 6 > 0 such that for each element x E Bx and each sequence (x,) in Bx with inf{()xm x,(( : m # n ) 2 E there is an index k for which We shall say that X has property w(P) provided it satisfies the condition obtained from the above one by replacing "for every E > O" by "for some E E (0,l)". In [6] it was observed that spaces with property w(P) are reflexive. Spaces dual to those with property (P) satisfy so called property (P*), which is known to imply NUS (see [6]). The same argument shows in fact that if a space X has property w(P), then X* is WNUS. We shall prove that the space X is WNUS too. Let (x,) be a basic sequence in the unit ball Bx of a space X with property w(P). Since X is reflexive, the sequence (x,) converges weakly to zero (see [2] p. 67). From the definition of property w(P) we obtain some constants E E (0 , l ) and 6 > 0. If llxnll > E for all n > 1, then passing to a subsequence, we can assume that inf{(lxm x,(l : m # n ) 2 E. Therefore there is an index k > 1 such that (1x1 + xkll 5 2(1 6) If llxkll 5 E for some k > 1, then In light of Proposition 3 this shows that the space X is WNUS. Since for reflexive spaces the weak Opial property is selfdual, we thus obtain the next corollary of Theorem 4. 610 D. KUTZAROVA, S. PRUS AND B. SIMS Corollary 6. If a Banach space X has property w(P) and the weak Opial property, then X and X * are orthogonally convex. Let us point out that the assumption of having the weak Opial property is essential in the above results. Example. There exists a NUS Banach space X which is not orthogonally convex. Let us consider the following norm in a two-dimensional space. where t = and a , p are real numbers. For an element x = (ai) E 12 we write S,X = (@,+I, a n + 2 , ...). Let now E denote the space 12 with the equivalent norm given by the formula where x = (a;) E 12 and 1 ) . is the initial norm of 12. I t is easy to see that for every x E E. Our space X is the Is-sum of countably many copies of E, so that X = 18{(Ej)), where Ei = E for all i. Therefore each element x E X is of the form x = (x(i)) with x(i) E E for all i. Let (en) be the natural basis of E. We consider elements e p E X such that e r ( i ) = 0 if i # m and e r ( i ) = ek if i = m. The set {ep}m,k,l rearranged into a sequence (x,) by the formula x, = e p if m + k = n gives us a basis of the space X. Let (P,) be the sequence of the natural projections associated to this basis, so that Pnx = CjnZl a j x j for any x = Cz, a j x j E X. It is easy to see that the sequence (2,) converges weakly to zero in X. Moreover D[(x,)] = 1)(1, -1)110 = l.lt. REMARKS ON ORTHOGONAL CONVEXITY OF BANACH SPACES 611 We take scalars a , p such that 11 and put a = g t a + 1, b = =tp. Let us now fix an arbitrary X > 0. For any m, k with k > 1, straightforward computations show that 'Therefore IMx(eT,er)( 2 llzll = Il(a, b)llo > D[(x,)]. Consequently which proves that X is not orthogonally convex. In order to show that X is NUS let us consider elements x , y E X such that there exists n for which Pnx = x and Pny = 0. We shall check that Let us fix i. If x( i ) # 0, then there are sequences of scalars ( a j ) , ( p j ) and an integer n > 1 such that n ~ ( i ) = Cj=l ajej Y ( i ) = Cpn+l PjejTo estimate Jlx(i) + y(i)(12 it suffices to consider an expression 2 2 s = Il(rl,rm)llo + 0.01 CEm+l yj 1 where rj = aj if 1 5 j 5 n and 3;. = pj if j > n. In the case when n > m , by (6) we have s 5 llx(i)112 + o.olllY(i)ll; 5 llx(i)l12 + llY(i)l12 If n < m , then D. KUTZAROVA, S. PRUS AND B. SIMS