ON ψ-direct SUMS OF BANACH SPACES AND CONVEXITY

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1 J. Aust. Math. Soc. 75 (003, 43 4 ON ψ-direct SUMS OF BANACH SPACES AND CONVEXITY MIKIO KATO, KICHI-SUKE SAITO and TAKAYUKI TAMURA Dedicated to Maestro Ivry Gitlis on his 80th birthday with deep respect and affection (Received 8 April 00; revised 3 January 003 Communicated by A. Pryde Abstract Let X ; X ;:::;X N be Banach spaces and a continuous convex function with some appropriate conditions on a certain convex set in R N. Let.X X X N / be the direct sum of X ; X ;:::;X N equipped with the norm associated with. We characterize the strict, uniform, and locally uniform convexity of.x X X N / by means of the convex function. As an application these convexities are characterized for the `p;q -sum.x X X N / p;q ( < q p, q <, which includes the well-known facts for the `p-sum.x X X N / p in the case p = q. 000 Mathematics subect classification: primary 46B0, 46B99, 6A5, 5A, 90C5. Keywords and phrases: absolute norm, convex function, direct sum of Banach spaces, strictly convex space, uniformly convex space, locally uniformly convex space.. Introduction and preliminaries A norm on C N is called absolute if.z ;:::;z N / =. z ;:::; z N / for all.z ;:::;z N / C N,andnormalized if.; 0;:::;0/ = =.0;:::;0; / = (see for example [3, ]. In case of N =, according to Bonsall and Duncan [3] (see also [], for every absolute normalized norm on C there corresponds a unique continuous convex function on the unit interval [0; ] satisfying max{ t; t}.t/ The authors are supported in part by Grants-in-Aid for Scientific Research, Japan Society for the Promotion of Science. c 003 Australian Mathematical Society /03 $A:00 + 0:00 43

2 44 Mikio Kato, Kichi-Suke Saito and Takayuki Tamura [] under the equation.t/ =. t; t/. Recently in [] Saito, Kato and Takahashi presented the N-dimensional version of this fact, which states that for every absolute normalized norm on C N there corresponds a unique continuous convex function satisfying some appropriate conditions on the convex set { } N N = t =.t ;:::;t N / R N : t ; t 0 under the equation.t/ = ( N = t ; t ;:::;t N. For an arbitrary finite number of Banach spaces X ; X ;:::;X N, we define the -direct sum.x X X N / to be their direct sum equipped with the norm.x ; x ;:::;x N / =. x ; x ;::: ; x N / for x X ; where term in the right-hand side is the absolute normalized norm on C N with the corresponding convex function. This extends the notion of ` p-sum of Banach spaces. The aim of this paper is to characterize the strict, and uniform convexity of.x X X N /. The locally uniform convexity is also included. For the case N =, the first two have been recently proved in Takahashi-Kato-Saito [3]and Saito-Kato [0], respectively. However the proof of the uniform convexity for the -dimensional case given in [0] seems difficult to be extended to the N-dimensional case, though it is of independent interest as it is of real analytic nature and maybe useful for estimating the modulus of convexity. Our proof for the N-dimensional case is essentially different from that in [0]. As an application we shall consider the ` p;q -sum.x X X N / p;q and show that.x X X N / p;q is uniformly convex if and only if all X are uniformly convex, where < q p ; q <. The same is true for the strict and locally uniform convexity. These results include the well-known facts for the ` p-sum.x X X N / p as the case p = q. Let us recallsome definitions. A Banachspace X or its norm is called strictly convex if x = y =.x = y/ implies.x + y/= <. This is equivalent to the following statement: if x + y = x + y ; x = 0; y = 0, then x = ½y with some ½>0 (see for example [9, page 43], []. X is called uniformly convex provided for any ž (0 <ž< there exists Ž>0 such that whenever x y ž. x = y =, one has.x + y/= Ž, or equivalently, provided for any ž (0 <ž< one has Ž X.ž/ > 0, where Ž X is the modulus of convexity of X, thatis, Ž X.ž/ := inf{.x + y/= ; x y ž; x = y =}.0 ž /: We also have the following restatement: X is uniformly convex if and only if, whenever x n = y n =and.x n + y n /=, it follows that x n y n 0. X is called locally uniformly convex (see for example [9, 4] if for any x X with x =and =

3 [3] On -direct sums of Banach spaces and convexity 45 for any ž.0 <ž</ there exists Ž>0 such that if x y ž; y =, then.x + y/= Ž. Clearly the notion of locally uniform convexity is between those of uniform and strict convexities.. Absolute norms on C N and ψ-direct sums (X X X N ψ Let AN N denote the family of all absolute normalized norms on C N.Let N ={.s ; s ;:::;s N / R N : s + s + +s N ; s 0. /}: For any AN N define the function on N by (.s/ =. s s N ; s ;:::;s N / for s =.s ;:::;s N / N : Then is continuous and convex on N, and satisfies the following conditions: (A 0 (A (A (A N.0;:::;0/ =.; 0;:::;0/ = =.0;:::;0; / = ; ( s s N.s ;:::;s N /.s + +s N / N s ;:::; N i= i s ; i= i.s ;:::;s N /. s / ( 0; s s ;:::; s N s ;... ( s s N.s ;:::;s N /. s N / ;:::; ; 0 : s N s N Note that from (A 0 it follows that.s ;:::;s N / on N as is convex. Denote 9 N be the family of all continuous convex functions on N satisfying (A 0, (A, :::,(A N. Then the converse holds true: For any 9 N define ( N ( z i= i z /( N z i= i ;:::; z N /( N z i= i (.z ;:::;z N / = if.z ;:::;z N / =.0;:::;0/; 0 if.z ;:::;z N / =.0;:::;0/: Then AN N and satisfies (. Thus the families AN N and 9 N are in oneto-one correspondence under equation ( (Saito-Kato-Takahashi [, Theorem 4.]. The ` p-norms { { z p + + z N p } =p if p < ;.z ;:::;z N / p = max{ z ;:::; z N } if p =

4 46 Mikio Kato, Kichi-Suke Saito and Takayuki Tamura [4] are typical examples of absolute normalized norms, and for any AN N we have (3 ([, Lemma 3.], see also [3]. The functions corresponding to ` p-norms on C N are {( p =p N = p.s ;:::;s N / = s p + s N } + +s p if p < ; { max } N s = ; s ;:::;s N if p = for.s ;:::;s N / N. Let X ; X ;:::;X N be Banach spaces. Let 9 N and let be the corresponding norm in AN N.Let.X X X N / be the direct sum of X ; X ;:::;X N equipped with the norm (4.x ; x ;:::;x N / :=. x ; x ;:::; x N / for x X : As is it immediately seen,.x X X N / is a Banach space. EXAMPLE. Let q p, q <. We consider the Lorentz `p;q -norm z p;q = { N.q=p/ = z } q =q for z =.z ;:::;z N / C N,where{z } is the nonincreasing rearrangement of { z }, thatis,z z z N. (Note that in case of p < q, p;q is not a norm but a quasi-norm (see [6, Proposition ], [4, page 6]. Evidently p;q AN N and the corresponding convex function p;q is obtained by (5 p;q.s/ =. s s N ; s ;:::;s N / p;q (for s =.s ;:::;s N / N, that is, p;q = p;q.let.x X X N / p;q be the direct sum of Banach spaces X ; X ;:::;X N equipped with the norm.x ;:::;x N / p;q :=. x ;:::; x N / p;q ; wecallitthe` p;q -sum of X ; X ;:::;X N.Ifp = q the ` p;p -sum is the usual ` p-sum.x X X N / p. For some other examples of absolute norms on C N we refer the reader to [] (see also []. 3. Strict convexity of (X X X N ψ A function on N is called strictly convex if for any s; t N.s = t/ one has..s + t/=/ <..s/ +.t//=. For absolute norms on C N,wehave

5 [5] On -direct sums of Banach spaces and convexity 47 LEMMA 3. (Saito-Kato-Takahashi [, Theorem 4.]. Let 9 N. Then.C N ; / is strictly convex if and only if is strictly convex. The following lemma concerning the monotonicity property of the absolute norms on C N is useful in the sequel. LEMMA 3. (Saito-Kato-Takahashi [, Lemma 4.]. Let 9 N. Let z =.z ; :::;z N /, w =.w ;:::;w N / C N..i/ If z w for all, then z w..ii/ Let be strictly convex. If z w for all and z < w for some, then z < w. THEOREM 3.3. Let X ; X ;:::;X N be Banach spaces and let 9 N. Then.X X X N / is strictly convex if and only if X ; X ;:::;X N are strictly convex and is strictly convex. PROOF. Let.X X X N / be strictly convex. Then, each X and.c N ; / are strictly convex since they are isometrically imbedded into.x X X N /. According to Lemma 3., is strictly convex. Conversely, let each X and be strictly convex. Take arbitrary x =.x /, y =.y /, x = y, in.x X X N / with x = y =. Let first. x ;:::; x N / =. y ;:::; y N /. Then, if x + y =, = x + y =. x + y ;:::; x N + y N /. x + y ;:::; x N + y N / x + y = ; from which it follows that x + y = x + y for all by Lemma 3.. As each X is strictly convex, x = k y with k > 0. Since x = y,wehavek = and hence x = y for all, orx = y, which is a contradiction. Therefore we have x + y <. Let next. x ;:::; x N / =. y ;:::; y N /. Since is strictly convex,.c N ; / is strictly convex by Lemma 3.. Consequently we have as is desired. x + y =. x + y ;:::; x N + y N /. x + y ;:::; x N + y N / =. x ;:::; x N / +. y ;:::; y N / < ; Now we see that the function p;q in the above example is strictly convex if < q p, q <. We need the next lemma.

6 48 Mikio Kato, Kichi-Suke Saito and Takayuki Tamura [6] LEMMA 3.4 ([5]. Let {Þ }; {þ } R N and Þ 0; þ 0. Let{Þ }; {þ } be their non-increasing rearrangements, that is, Þ Þ Þ and N þ þ þ. Then N Þ N = þ N = Þ þ. PROPOSITION 3.5. Let < q p, q <. Then the function p;q given by (5 is strictly convex on N. PROOF. Let s =.s /; t =.t / N, s = t. Without loss of generality we may assume that Put.s + t /.s N + t N / s + t s N + t N 0: =. s s N ; =p =q s ;:::;N =p =q s N /; =. t t N ; =p =q t ;:::;N =p =q t N /: Then by Lemma 3.4 we have q = {. s s N / q + q=p s q + + N q=p s q N. s s N ; s ;:::;s N / p;q = p;q.s/ and q p;q.t/. On the other hand, p;q ( {( N q N s + t s i + t i ( } =q = +.i + / q=p si + t q i i= i= [( {( ( } N N q = s i + t i i= i= N ( { } ] =q q +.i + / =p =q s i +.i + / =p =q t i = i= } =q + : q Since `q-norm q ( < q < is strictly convex and s = t, wehave + q < q + q. Indeed, if + q = q + q, then = k with some k > 0 (note that = 0, = 0. Hence s = kt for all, and N s i= i = k ( N t i= i. Therefore,k = and we have s = t, which is a contradiction. Consequently, ( s + t p;q or p;q is strictly convex. = + < q + q q p;q.s/ + p;q.t/ ; By Theorem 3.3 and Proposition 3.5 we have the following result for the ` p;q -sum of Banach spaces.

7 [7] On -direct sums of Banach spaces and convexity 49 COROLLARY 3.6. Let < q p ; q <. Then, `p;q -sum.x X X N / p;q is strictly convex if and only if X ; X ;:::;X N are strictly convex. In particular, the ` p-sum.x X X N / p ; < p <, is strictly convex if and only if X ; X ;:::;X N are strictly convex. 4. Uniform convexity of (X X X N ψ Let us characterize the uniform convexity of.x X X N /. THEOREM 4.. Let X ; X ;:::;X N be Banach spaces and let 9 N. Then.X X X N / is uniformly convex if and only if X ; X ;:::;X N are uniformly convex and is strictly convex. PROOF. The necessity assertion is proved in the same way as the proof of Theorem 3.3. Assume that X ; X ;:::;X N are uniformly convex and is strictly convex. Take an arbitrary ž>0 and put Ž := Ž X.ž/ = inf{ x + y : x y ž; x = y = }: We show that Ž>0. There exist sequences {x n } and {y n } in.x X X N / so that (6 and (7 x n y n ž; x n = y n = lim n x n + y n = Ž: Let x n =. ;:::; N / and y n =.y.n/ ;:::;y.n/ N /. Since for each N, =.0;:::;0; ; 0;::: ;0/ x n = and y.n/ y n = for all n, the sequences { } n and { y.n/ } n have a convergent subsequence respectively. So we may assume that a, y.n/ b as n. Further, in the same way, we may assume that (8 and (9 y.n/ c as n + y.n/ d as n : Put K n = N.n/ = x. Then x n = K n. =K n ;:::; N =K n / =. Letting n,as is continuous, we have (0 ( N ( a.a ;:::;a N / = a N a ;:::; = = a N N = a = :

8 40 Mikio Kato, Kichi-Suke Saito and Takayuki Tamura [8] Also we have ( ( N ( b b N.b ; ; b N / = b N b ;:::; N = = : = = Next let n in (6, or in ( N x n y n = y.n/ = ( y.n/ N = y.n/ ;::: ; N y.n/ N N ž: = y.n/ Then we have ( N (.c ;:::;c N / = = ( c c N c ;:::; = by (8. In the same way, according to (7 and (9, we have c N N = c ž (3.d ;:::;d N / = Ž: Now, assume that.a ;:::;a N / =.b ;:::;b N /. Then, according to (0, ( and the strict convexity of we obtain that Ž =.d ;:::;d N /.a + b ;:::;a N + b N / < ; which implies Ž>0. Next, let.a ;:::;a N / =.b ;:::;b N /. Since.c ;:::;c N / =.0;:::;0/ from (, we may assume that c > 0 without loss of generality. Then as ( c = lim y.n/ lim + y.n/ = a + b = a ; n n we have a > 0and 0 < c = lim a n y.n/ = lim n y.n/ (4 y.n/ : Indeed, we have the latter identity because y.n/ y.n/ y.n/ y.n/ y.n/ b a b = 0 as n :

9 [9] On -direct sums of Banach spaces and convexity 4 Since X is uniform convex, it follows from (4 that d = lim a n + y.n/ = lim n + y.n/ y.n/ < ; whence d < a. Accordingly, by (3 and Lemma 3. we obtain that Ž =.d ; d ;:::;d N / <.a ; a + b ;:::;a N + b N / =.a + b ; a + b ;:::;a N + b N /.a ;:::;a N / +.b ;:::;b N / = ; which implies Ž>0. This completes the proof. The parallel argument works for the locally uniform convexity and we obtain the next result. THEOREM 4.. Let 9 N. Then.X X X N / is locally uniformly convex if and only if X ; X ;:::;X N are locally uniformly convex and is strictly convex. Indeed, for the sufficiency, take an arbitrary x.x X X N / with x = and merely let x n = x in the above proof. By Theorem 4. and Theorem 4. combined with Proposition 3.5 we obtain the following corollary. COROLLARY 4.3. Let < q p, q <. Then, `p;q -sum.x X X N / p;q is uniformly convex (locally uniformly convex if and only if X ; X ;:::;X N are uniformly convex (locally uniformly convex. In particular, the ` p-sum.x X X N / p ; < p <, is uniformly convex (locally uniformly convex if and only if X ; X ;:::;X N are uniformly convex (locally uniformly convex. References [] B. Beauzamy, Introduction to Banach spaces and their geometry, nd edition (North-Holland, Amsterdam, 985. [] R. Bhatia, Matrix analysis (Springer, Berlin, 997. [3] F. F. Bonsall and J. Duncan, Numerical ranges II, London Math. Soc. Lecture Note Ser. 0 (Cambridge University Press, Cambridge, 973. [4] J. Diestel, Geometry of Banach spaces, Lecture Notes in Math. 485 (Springer, Berlin, 975.

10 4 Mikio Kato, Kichi-Suke Saito and Takayuki Tamura [0] [5] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities (Cambridge University Press, Cambridge, 967. [6] M. Kato, On Lorentz spaces `p;q {E}, Hiroshima Math. J. 6 (976, [7] D. Kutzarova and T. Landes, Nearly uniform convexity of infinite direct sums, Indiana Univ. Math. J. 4 (99, [8], NUC and related properties of finite direct sums, Boll. Un. Math. Ital. (7 8A (994, [9] R. E. Megginson, An introduction to Banach space theory (Springer, New York, 998. [0] K.-S. Saito and M. Kato, Uniform convexity of -direct sums of Banach spaces, J. Math. Anal. Appl. 77 (003,. [] K.-S. Saito, M. Kato and Y. Takahashi, On absolute norms on C n, J. Math. Anal. Appl. 5 (000, [], von Neumann-Jordan constant of absolute normalized norms on C, J. Math. Anal. Appl. 44 (000, [3] Y. Takahashi, M. Kato and K.-S. Saito, Strict convexity of absolute norms on C and direct sums of Banach spaces, J. Inequal. Appl. 7 (00, [4] H. Triebel, Interpolation theory, function spaces, differential operators (North-Holland, Amsterdam, 978. Department of Mathematics Kyushu Institute of Technology Kitakyushu Japan katom@tobata.isc.kyutech.ac.p Department of Mathematics Faculty of Science Niigata University Niigata Japan saito@math.sc.niigata-u.ac.p Graduate School of Social Sciences and Humanities Chiba University Chiba Japan tamura@le.chiba-u.ac.p