Barrelledness of Spaces with Toeplitz Decompositions

Journal of Mathematical Analysis and Applications 240, 468-480 (1999) Article ID jmaa.1999.6603, available online at http://www.idealibrary.com on IOErl;L@ Barrelledness of Spaces with Toeplitz Decompositions' Pedro J. Pafil, Carmen SAez, and Juan M. Viruks Escuelu Superior de Ingenievos, Univevsidud de Sevillu, Cumino de 10s Descubnmientos s/n, 41092-Seville, Spain E-mail: [email protected]; [email protected] Submitted by John Hovva'th Received May 14, 1998 DEDICAMOS ESTE ARTfCULO CON TODO CARIRO A NUESTRO MAESTRO Y AMIGO MIGUEL FLORENCIO LORA A Toeplitz decomposition of a locally convex space E into subspaces (E,) with projections (P,) is a decomposition of every x E E as x = C,P,x, where ordinary summability has been replaced by summability with respect to an infinite and lower triangular regular matrix. We extend to the setting of Toeplitz decompositions a couple of results about barrelledness of Schauder decompositions. The first result, given for Schauder decompositions by No11 and Stadler, links the barrelledness of a normed space E to the barrelledness of the pieces E, via the fact that E' is big enough so as to coincide with its summability dual. Our second theorem, given for Schauder decompositions by Diaz and Miiiarro, links the quasibarrelledness of an No-quasibarrelled (in particular, (DF)) space E to the quasibarrelledness of the pieces Ek via the fact that the decomposition is simple. Key Wovds: decompositions of locally convex spaces; barrelledness; summability and bases; (DF)-spaces; sequence spaces. o 1999 Academic Press INTRODUCTION Up to what point can one substitute ordinary summability by a matrix summability method in the definition of a Schauder decomposition, and, still, obtain nice results about the locally convex structure of the space in terms of the locally convex structure of its pieces? Our purpose here is to Research partially supported by la Consejerfa de Educaci6n y Ciencia de la Junta de Andalucia and by la Direcci6n General de Investigaci6n Cientifica y TCnica, project PB941460. 4 68 0022-247)3/99 $30.00 Copyright 0 1999 by Academic Press All rights of reproduction in any form reserved. BARRELLEDNESS AND TOEPLITZ DECOMPOSITIONS 469 extend two theorems about the barrelledness of certain locally convex spaces having a Schauder decomposition-obtained, respectively, by No11 and Stadler [14] (main hypothesis: a normed space with a shrinking decomposition) and Diaz and Miiiarro [5] (main hypothesis: the space is (DF) and the decomposition is equicontinuous)-to the setting of decompositions defined in terms of more general matrix summability methods. In a similar framework (scalar sequence spaces having a generalized sectional convergence scheme), several interesting barrelledness results have been given by Ruckle and Saxon [17]. Although we refer the reader to this very interesting paper, we shall make some comments about it at the end of our work. Although our notation and terminology will be mostly standard, e.g., cp is the space of finitely non-zero sequences, c is the space of convergent sequences, eLk1 stands for the kth unit sequence (we refer the reader to [8, 15, 16, 18, 22]), let us recall a few facts from summability theory. Let T = [tnk] be an infinite matrix of scalars from the field K of real or complex numbers. The matrix T is said to be row-finite if each row of T is in cp, an Sp,-matrix if each column of T is convergent to 1, and reversible if for every sequence y E c the infinite system of linear equations T. x = y has unique solution. It is well known [22, 5.4.555.4.91 that each row-finite and reversible T has a unique two-sided inverse matrix Tpl such that each row of Tpl is in 1' and for each y E c the unique solution of T.x = y is Tpl 'y. An important particular case is that of a triangle. Following Wilansky [22], a lower triangular infinite matrix with non-zero diagonal entries is called a triangle. A triangle is always row-finite and reversible, and its inverse is also a triangle. Let E be a locally convex space. The convergence field of T in E is the space c,(E) of all sequences (xk) from E such that the product T. (xk) is a convergent sequence in E. The sequences in c,(E) are said to be T-convergent. For (xk) E c,(E) the limit of the sequence T. (xk) is called the T-limit of (xk) and will be denoted by T-lim xk, in other words Terminology and Notation. We simply denote by c, the convergence field of T in K. If T is a row-finite and reversible matrix then the norm llxll~ := IIT.xllm makes c, a Banach space (isomorphic to c). Let T = [tnk] be a row-finite infinite matrix of scalars. A sequence (pk) of non-trivial, mutually orthogonal, and continuous linear projections defined on a locally convex space E is said to be a Toeplitz decomposition of E with respect to the matrix T or, for short, a T-decomDEFINITIONS. 470 PA~L, SAEZ, AND VIRUBS position of E, if x = T-lim Pkx Alternatively, if we define the sequence of operators for every x E E. T,: X EE + T,(X) := xt,kpkx EE, k then (Pk) is a T-decomposition of E whenever limn T,x = x for every x E E. If 2 stands for the triangle with all of its lower triangular entries equal to 1, then 2-decompositions are the familiar Schauder decompositions. It is important to note that the operators T, are not projections in general (they are increasing projections in the case of a Schauder decomposition), however we do have T,P, = P,T, = tnkPk for all n, k E N. Note also that the sequence of operators (T,) is precisely the product T. (Pk). Hence saying that limn T,x = x is the same as saying that the sequence T. (P,x) converges to x. Call Ek := P,(E). Since E, does not reduce to the zero subspace and for every xk E E, we have xk = limn Tnxk = limn tnkXk, it follows that limn t,, = 1, i.e., T is an Sp,-matrix. Let us also point out, for later use, that, as is easy to see, we can write T,(E) = En, @ triangle then T,(E) is a complemented subspace of El @ E, @ ... @ En. Still another way of looking at a Toeplitz decomposition is the following: Every E, is a complemented subspace of E and we can identify every x E E with the vector-valued sequence (Pkx) E nE,, so that E becomes a linear subspace of nE, that, with the topology translated from E, has the set of all finite sequences as a dense subspace because limn T,x = x for every x E E and T is row-finite. We extend now some of the terminology commonly used for Schauder decompositions (see [9, 10, 18, 20, 211): A T-decomposition (Pk) of a locally convex space E is said to be finite-dimensional if every E, is finite-dimensional, equicontinuous if the sequence of operators (T,) is equicontinuous, and complete if for each sequence (x,) E nE, such that the product T. (x,) is a Cauchy sequence in E there exists x E E such that xk = Pkx for every k E N, and a fortiori, T. (xk) converges to x. Using primes to denote adjoint operators, for every x E E and every u E E' we can write En, @ ... , where En, = Ek if t,, # 0 and En, = {o} if t,, = 0. If T is a BARRELLEDNESS AND TOEPLITZ DECOMPOSITIONS 47 1 This shows that (PL) is also a T-decomposition of E’ endowed with the weak topology a(E‘, E). If we call E, := P,(E) and EL := PL(E’) then the dual of E, can be identified with EL. The computation above also shows that the sequence (T,’u) is a(E‘, E)-bounded. A T-decomposition (P,) of a locally convex space E is said to be shrinking if (PL) is also a T-decomposition of E‘ endowed with the strong topology P(E’, E) and is said to be simple if (Tiu) is a P(E’, E)-bounded sequence for every u E E’. A K-space is a locally convex sequence space h 3 cp such that the kth projection defined by n-,((x,),) := x,e[,] is continuous for every k E N. A K-space h is said to have property T-AK if x = T-lim xkeLkl for every sequence x = (x,) E A. Thus, a sequence space h has property T-AK if and only if the sequence (n-,) is a (one-dimensional) T-decomposition of h or, in other words, the sequence of operators defined by 7, := Cktnkn-,, i.e., (7,) := T. (n-,), satisfies x = lim, ~,(x,) for every sequence x = (x,) E h (see [3, 4, 121). (When dealing with scalar sequence spaces, we shall keep the notations (n-,) and (7,) throughout the paper). In particular, h has the property 2-AK means precisely that (eLkl) is a Schauder basis of A. The matrices T such that cT has property T-AK where characterized by BUNTINAS’S THEOREM. Let T be a row-finite and reversible Sp,-matrix. (1) The sequence of coordinate projections (n-,) is a T-decomposition (2) The sequence of operators (7,) is equicontinuous on cT. (3) If we denote Tpl by [s,,] then EXAMPLE. Buntinas [4, Theorems 8-10]. Then the following conditions are equivalent: Of cT’ (4) The dual (c,)’ can be identifed with the multiplier space (cT + cT) formed by the sequences y such that the coordinatewise product xy is in cT for every x E cT and, in this case, the bilinear form of the dual pair is given by (X? Y)(CT,(CT)’) = T-lim xy. The first non-trivial examples of matrices T such that cT has property T-AK are the series-to-sequence Ceshro matrices of order ctl 2 0; this was proved by Zeller [23]. Therefore, to avoid clumsy repetitions, a row-finite 472 PA~L, SAEZ, AND VIRUBS and reversible Sp,-matrix T such that cT has property T-AK will be called a Zeller-Buntinas matrix. Note that if T is a Zeller-Buntinas matrix then cT is a sum space in the sense of Ruckle [16]. BARRELLEDNESS OF NORMED SPACES WITH SHRINKING TOEPLITZ DECOMPOSITIONS Our first purpose is to extend to the setting of Toeplitz decompositions with respect to triangles a result due to No11 and Stadler [14] that links the barrelledness of a normed space E to the barrelledness of the pieces E, via the fact that E' is big enough. DEFINITION [14]. Let E be a locally convex space having a T-decomposition (P,). The &dual of E is the space EOT defined by any of the equivalent formulations i cc Ep~ := ( (u,) E n EL : ((x, u,)), is T-convergent for all x E E k= 1 Every (uk) E EOT defines a linear form in E, namely, x E E + limnCktnk(x, uk) and, in this sense, E' c EpT. It is clear that the equality E' = Ep~ holds if and only if (PL) is a complete T-decomposition of E' endowed with the weak topology a(E', E). Let E[ II .I11 be normed space such that E[ a(E, E')] has a shrinking Toeplitz decomposition (Pk) with respect to a triangle T. Then (Pk) is an equicontinuous Toeplitz decomposition of E[ll . 111 and the following conditions are equivalent: THEOREM 1. (i) E is barrelled; (ii) E' = EpT and E, is barrelled for every k E N. Pro05 As is well known, every projection Pk is norm-continuous because it is a(E, El)-continuous. Now, since (P,) is shrinking and E' is a Banach space, it follows that the sequence of operators (T,') is normequicontinuous in E' and a standard duality argument shows that (T,) is also norm-equicontinuous in E. Finally, since the linear span of the subspaces (E,) is dense in E[ 11 and for every xk E E, the convergence to xk of the sequence (Tnxk), holds in the norm topology (because it reduces to the convergence to 1 of the columns of T), a standard equicontinuity argument originally due to Mazur (see [ 11, Sect. 39.4.( l)] or [7]) shows that (T,x) converges to x in the norm-topology for every x E E. BARRELLEDNESS AND TOEPLITZ DECOMPOSITIONS 473 That (i) implies (ii) is an easy consequence of the Banach-Steinhaus theorem. To prove that (ii) implies (i), let us carefully refine No11 and Stadler's original proof. Let us begin by noting that for each n E N we have that T'(E) is a barrelled space because, as we pointed out above, T'(E) is a complemented subspace of El @ E, @ ... @ En. Proceed now by contradiction and assume that there is a a(E', E)-bounded sequence (0,) such that Ilu,ll 2 m2" for each m E N. Take ym := m-lu, for m = 1,2,. . . to define a sequence (y,) that a(E', E)-converges to zero but such that lly,ll 2 2" for each m E N. We shall prove by induction that there are a couple of increasing sequences (mj) and (nj) from N such that for every j E N, ~~y,-T'y,~l~2~J forallm=1,2, ..., mj-,anda1ln2nj, (1) (2) Assume that mj and nj have been found such that (1) and (2) hold. Since (P,) is shrinking, we have that limJy, - T',y,ll = 0 for all m = 1,2,. . . , mj so there is an index nj+l > nj such that (1) holds with j replaced by j + 1. On the other hand, for each n = 1,2,. . . , nj+ the sequence (T'u,), is a(T',(E'), T'(E))-bounded in the dual T'(E') of the barrelled space T'(E). Therefore, each of the sequences (T'u,), (n = 1,2,. . . , nj+>) is norm-bounded and, since llT'y,ll = m-lllT'u,ll, it follows that there is an index mj+ such that (2) also holds with j replaced by j + 1. I 2-9. Now fix n E N and take j, such that n I njo. Inequality (2) yields ~~T'y,~~ I 2-J for all n = 1,2,. . . , nj and all m 2 mj. For each j E N call aj := IIy m, c ajllT'ym,Il I 2-Jo c 2-", < 00 j2jo j2jo so that the series Cy= ajT'y,, converges in the Banach space T'(E') to some element 2,. We shall prove now that (2,) is a a(E',E)-Cauchy sequence. Fix x E E with llxll I 1 and for each n take j =j(n) such that nj(') I n < nj(')+ 1. Then we may write jh1 c3 (XJ,) = c aj(x,T'Ym,) + aj(n)(X?T'Ym,(,,) + c aj(x?T'Ym,). j= 1 j=j(n)+ 1 (3) Let us see that the three summands in the right hand side converge as n + 00. The first one can be written as 474 PA~L, SAEZ, AND VIRUBS I c aJ(x,T,'y,,) j=j(n)+ 1 The first term in the right side of equality (4) converges because (y,) is a a(E', E)-bounded sequence and aJ I 2-"1 for each j E N. The second term converges because inequality (1) tells us that I c 2Tm12-J(n) j=j(n)+ 1 This shows that the first term in equality (3) converges. The central term in equality (3) can be written as aJ(n)(x9 T,lYmIcn)) = aj(n)(Tnx, Ym,(,) ) - - (Tnx - '9 aj(n)Yrn,(,)) + (x, aj(n)Yml(n)), where the first summand converges to zero because ~~aJ~n~yml~n~~~ = 1 and T,x converges to x in the norm, and the second summand also converges to zero because (y,) is a(E', E)-convergent to zero and aJ I 2-"1. This shows that the central term in equality (3) also converges. Finally inequality (2) tells us that Icc I cc tends to zero as n + 00. This shows that the third summand in the right side of (3) also converges. Define now the sequence (uk) := TP1 . (2,) (recall that TP1 is also a triangle). We shall prove that (uk) is in E". Denote the entries of the matrix TP1 by skn so that uk = Cnsknz,. To see that uk is in EL it is enough to check that if i # k then Pi'uk = 0. Put Pi'uk = CnSknPi'Z,. NOW, since the series CJaJT,'y,, is norm-convergent to z,, since the sum in n is finite, and since P,'T,' = t,,P,' for every i, n E N because the projections (P;) are mutually orthogonal, we have On the other hand, T. (uk) = T. TP1 . (2,) = (2,) that, as we have seen, is a a(E', E)-Cauchy sequence. This shows that (uk) E E". By our hypothesis, there exists some element u E E' such that uk = PLu for every BARRELLEDNESS AND TOEPLITZ DECOMPOSITIONS 475 k E N. Since the decomposition is shrinking, we have that (2,) = T. (uk) = T. (PLu) is norm-convergent to u, and In norm the first and last terms in the right side do not exceed 2 -J C, a, I 2-J by (1) and (2), and thus tend to zero. Therefore, u = lim, z, = lim, C;:;a,ymz, but this is a contradiction because lla,ymzIl = 1. Let E be normed space such that E[a(E, E‘)] has a finite-dimensional shrinking Toeplitz decomposition (Pk) with respect to a triangle T. Then (Pk) is an equicontinuous Toeplitz decomposition of E and the following conditions are equivalent: I COROLLARY 1.1. (i) E is barrelled; (ii) E‘ = E”. Remarks. (1) The natural hypothesis E‘ = Ep~ is itself a sort of weak barrelledness condition. It can be proved that if T is a Zeller-Buntinas triangle and (P,) is a T-decomposition of a locally convex space E with barrelled subspaces E,, then E’ = EOT if and only if a barrel U c E is a zero-neighborhood for the Mackey topology in E provided that T, converges pointwise to the identity for the normed topology generated by the gauge of U. (2) The hypothesis that the space is normed cannot be suppressed in general. For instance, co[ a(co, 11)] satisfies all the hypotheses of Theorem 1 except that of being normed and, obviously, it is not barrelled. Metrizability is not enough, either, because the sequence space E := {x E o : sup, nixzn - xZn+ lI < m$ is not a barrelled subspace of o but EOT = E‘ = cp [l]. We do not know if Theorem 1 holds for (DF)-spaces. The strong dual of a (DF)-space is a Frkchet space and one could try to adapt Stadler and Noll’s proof to this case, but the crucial step of passing from (v,) to (y,) cannot be done in a non-normable Frkchet space. More concretely, a Frkchet space E turns out to be a Banach space if and only if whenever (v,) is an unbounded sequence from E, there is an unbounded subsequence (vnk) and a sequence of scalars (a,) convergent to zero such that ( a,v,,) is bounded but does not converge to zero. (3) Within the framework of scalar sequence spaces and ordinary convergence, the class of spaces to which Theorem 1 applies resembles the spaces with the so-called Wilansky property [l, 141. This concept has been 476 PA~L, SAEZ, AND VIRUBS extended to T-convergence by No11 [13]: Let T be an Sp, triangle, an FK-space h containing cp is said to have the Wilanshy (&W) property if whenever p is a subspace of h such that = AOT then p is barrelled. Theorem 1 implies that if h is a BK-space such that both h and A’ have property T-AK for a triangle T, then h has property ( &W), a result that can also be deduced from Noll’s results in [ 131. More generally we have the following result. Let (Pk) be a shrinking Toeplitz decomposition of a Banach space E with respect to a triangle T. Let F be a subspace of E containing all subspaces (E,). Then F is barrelled if and only if FOT = E”. COROLLARY 1.2. BARRELLEDNESS OF (DF)-SPACES WITH TOEPLITZ DECOMPOSITIONS Our second purpose is to extend to the setting of Toeplitz decompositions a result due to Diaz and Miiiarro [5] that links the quasibarrelledness of a (DF)-space E to the quasibarrelledness of the pieces E, via the fact that the decomposition is equicontinuous. This result was motivated by the problem of the stability in tensor products of Frkchet spaces of some topological vector space properties (see also [2]). Let E be an KO-quasibarrelled space with a simple Toeplitz decomposition (Pk) with respect to a row-finite matrix T. Then the decomposition is equicontinuous. Moreover, if every E, is quasibarrelled then E is quasibarrelled. We start by proving that the decomposition is equicontinuous. According to [8, 12.2.11, it will be enough to prove that if B is a bounded subset of E then U, T,(B) is also bounded. But this follows from THEOREM 2. Pro05 and the latter supremum is finite because, since the decomposition is simple, we know that (Tiu) is a P(E’, E)-bounded sequence for every u E E’. Let now W be a bornivorous barrel in E. We must prove that W is a zero-neighborhood. Each T,( E) is quasibarrelled, being complemented in a quasibarrelled space of the form El cI3 ... cI3 E,, so Wn T,(E) is a zero-neighborhood in T,(E), and TL1(W) = TL1(W n T,(E)) is a barrel and a zero-neighborhood in E. Hence, by hypothesis, U := 0, TnP1(W) is a zero-neighborhood in E, provided U is bornivorous. Moreover, U c W