Absolute Boundedness and Absolute Convergence in Sequence Spaces

proceedings of the american mathematical society Volume 111, Number 4, April 1991

ABSOLUTE BOUNDEDNESS AND ABSOLUTE CONVERGENCE IN SEQUENCE SPACES

MARTIN BUNTINAS AND NAZA TANOVIC-MILLER

(Communicated by J. Marshall Ash)

Abstract. Let ft* be the set of all sequences h = (hk)k*Axof Os and Is. A sequence x in a topological sequence space E has the property of absolute boundedness \AB\ if ft* • x = {y\yk = hkxk , h € ft*} is a bounded subset of E . The subspace E,AB, of all sequences with absolute boundedness in E has a natural topology stronger than that induced by E. A sequence x has the property of absolute sectional convergence \AK\ if, under this stronger topology, the net {h • x} converges to x , where h ranges over all sequences in ft* with a finite number of Is ordered coordinatewise (h1 < h" iff V/c, hk < hk ). Absolute boundedness and absolute convergence are investigated. It is shown that, for an F.K-space E, we have E = E,AB, if and only if E = l°° • E, and every element of E has the property \AK\ if and only if E = c0 • E . Solid hulls and largest solid subspaces of sequence spaces are also considered. The results are applied to standard sequence spaces, convergence fields of matrix methods, classical Banach spaces of Fourier series and to more recently introduced spaces of absolutely and strongly convergent Fourier series.

1. Introduction We mainly use standard notation as given in §2. For an FF-space F, various forms of sectional boundedness and sectional convergence have been shown to be equivalent to invariances of the form F = D- E with respect to coordinatewise multiplication by some space D. Such statements show the equivalence of topological properties of F with algebraic properties of F. In 1968 Garling [9] showed that an FF-space F has the property of sectional boundedness AB if and only if F is invariant with respect to the space bv of sequences of bounded variation, and that F has the property of sectional convergence AK if and only if F = bvQ• F. In 1970 Buntinas [4] showed that, for an FF-space F, Cesàro sectional boundedness oB is equivalent to invariance with respect to the space q of bounded quasiconvex sequences and that Cesàro sectional convergence oK is equivalent to invariance with respect to the space q0 = q n c0 of quasiconvex null sequences. In 1973 results were obtained for more general Toeplitz sections Received by the editors July 17, 1989 and, in revised form, January 18, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 46A45; Secondary 42A16, 42A28. Research partially supported by U.S.-Yugoslav Joint Fund (NSF JF 803).

967

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[5]. In 1977, Sember [12] and Sember and Raphael [13] showed that for FK- spaces, unrestricted sectional boundedness UAB is equivalent to invariance with respect to the space c of convergent sequences and that unrestricted sectional convergence UAK is equivalent to invariance with respect to the space c0 of null sequences. In this paper we study absolute boundedness \AB\ and absolute convergence \AK\. These conditions are stronger than UAB and UAK, respectively. How- ever, for FF-spaces, we show that the property \AK\ is equivalent to UAK . Among other results, we show in §3 that an FF-space F has absolute boundedness if and only if it is solid (A-invariant) and that it has absolute convergence if and only if it is c0-invariant. The intersection of all solid FK containing an FF-space F is called the solid hull of F. In §4 we show that it is an FF-space and characterize it as an FF-product space. We show that the solid hull of an FF-space F is related by duality to the \AB\ subspace of F. In the last section, we give examples and applications to summability theory and Fourier analysis.

2. DEFINITIONS Let co be the space of all real or complex sequences x = (xk). An FF- space is a subspace of co with a complete metrizable locally convex topology with continuous coordinate functional fk : x —►xk for all k . An FF-space whose topology is defined by a norm is a Banach space and is called a FF- space. Let e be the sequence with 1 in the kth coordinate and 0 elsewhere, and let cp be the linear span of {e 12, e , e 3 ,...}. In this paper we consider only FK- and FF-spaces containing tp, although all the definitions apply to more general F-spaces containing subsets A and B of co, we use A • B := {x -y\x e A, y e B}. If D c co and F is an FF-space, we define the F-dual of D as the multiplier space D = (D —>F) := {y e co\x ■y e F for all x e D}. Let s" := ELie* = (1 A, ... , 1,0,...) and let e:= (1, 1, 1, ...) be the sequence of all "ones." The «th section of a sequence x is snx := s" • x = (Xj , x2, ... , Xn , U , . . . ) . A sequence x in co has the property AB of sectional boundedness in an FF-space F if the sections s" x of x form a bounded subset of F, and it has the property AK of sectional convergence if, in addition, the sections converge to x in the topology of F. Let / := {A s co\hk = 1 or hk = 0 for all k} and ^ := ¿F n

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F and we say that x has the property \AB\ of absolute boundedness if MA• x is a bounded subset of F . For each FF-space F, we define the space EAB consisting of all elements x of co with the property AB in F. Similarly, for the properties UAB and \AB\, we obtain spaces EUAB and E,AB, . That is, EAB = {x e co\{s"x}™=xis a bounded subset of F} , EUAB= {x e co\Mf • x is a bounded subset of F}, F|^B| = {x e co\M"• x is a bounded subset of F} . Each of these spaces is an FF-space under an appropriate topology discussed in §3. These spaces are not necessarily subspaces of F, as is shown by the example (c0)UAB= (c0)AB= l°° . However, E.AB, is always a subspace of F, since e e MA. We say that an FF-space F has the property AB, UAB, or \AB\ if F is a subset of EAB, EUAB, or E,AB,, respectively. Clearly we have E:AB, c EUABc EAB. The converse inclusions do not generally hold. For example, the FF-space c of all convergent sequences has the property UAB but it does not have the property \AB\. The set Mf is a directed set under the relation h" > h' defined by h'k > h'k for all F. A sequence x in an FF-space F containing cp has the property UAK in F if the net h • x, where h ranges over MA, converges to x under the topology of F. We say that x has the property \AK\ of absolute sectional convergence if MA'• x c F and the net h • tí • x , where h ranges over MA, converges to tí • x uniformly in tí e MA under the topology of F. We define EAK to be the space of all elements x of F with the property AK in F. The same can be done for the properties UAK and \AK\. That is, EAK = {x e E\ lim^ snx = x} , EUAK= {x e E\ limAh • x = x, h e M^} , EiAKi = {x e E\ lim^ h ■tí • x = tí ■x, uniformly in h' e MA, for he*,}. The space EAD is the closure of cp in F. Since cpc F, we have the inclusions 9 c E\AK\ C Euak c &AK c EAD C F. If EAD = F, we say that F has the property of sectional density AD. If y e E whenever \yk\ < \xk\ for some x e E, we say that F is solid; this is equivalent to /°°-invariance: F = l°° -E. We finish this section with a list of some FF-spaces and their norms. The FF-spaces l°° , c, and c0 are the space of all bounded, convergent, and null sequences x, respectively, under the sup norm H-xll^ := sup¿. \xk\ ; bv is the FF-space of all sequences x of bounded variation under the norm \\x\\bv := Z)fcli \xk ~ xk+\\ + Hxlloo' bv0 = bv n c0 under the same norm; cs is the FF-space of sequences x with convergent series under the norm \\x\\bs := SUP„I Yll=i xk\' IP ' f°r 1 < P < oo , are the FF-spaces of sequences x with absolutely p-summable series under the norm ||x|| := (J2T=i \xk\P) > tne mixed lp'q spaces (1 < p < oo, 1 < q < oo) [11] consist of all x with

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11*11,,,:= (E%i(\\dJx\\P)P)1/9 < °° where dJxk = xk for 1' < k < lj+l and dJxk = 0 elsewhere; for q = oo , ||x||p q := supy ||iFx|| . Clearly lp'p = lp .

3. Absolute boundedness and absolute convergence The properties UAB and UAK were investigated by Sember [12] and Sem- ber and Raphael [13]. The properties \AB\ and \AK\ considered here are related. Let F be a FF-space under the norm ||x||£ . We define the (extended) absolute norm on F by IWIi£l:= sup||Ax||£,

with the convention that \\h ■x\\E = oo whenever h ■x £ E. Clearly, ||x||£ < ||x|||£| and E]ABl= {x € F|||x|||£| < oo}. Similary, if F be an FF-space with an increasing family of seminorms P 1 < P 2 <" P ^ < • • • defining the topology of F, we define the (extended) absolute seminorms by plEAx) := sup p (h ■x), k=l,l,3,...,

with the convention that p (h ■x) = oo whenever h • x $ E . Clearly, E,AB, = {x e E\pkE¡(x) < oo for k = 1,2,...}. By Garling's Theorem [9, p. 998], E,AB, is an FF-space under the seminorms pl, /A , p , p,E,, ... . Since p < p,E, for all k , px, p2, p , ... may be omitted. Hence the following theorem and corollary hold: Theorem 1. Let E be an FK-space with defining seminorms pl, p2, p3, ... . Then E,AB, is an FK-space whose topology is defined by the seminorms p!E,, 2 3 P\E\ ' P\E\. Corollary. If E is a BK-space under the norm || • ||£, then E,AB, is a BK-space under the norm || • ||,£,. Remark. It follows that a sequence x in an FF-space F has the property \AK\ if and only if the net A? • x converges to x in the topology of E,AB, . Theorem 2. Let E bean FK-space containing cp. Then the following statements are equivalent: [a] x e F|^B| ; [b] M*-xcE;and [c] r-xcE. Proof. We have [a] => [b] by definition of \AB\. Suppose [b]. Then MAc ({x} —>F). The multiplier space ({x} —»F) is an FF-space [10, p. 229]. By Bennett and Kalton [3], MA is a subset of an FF-space if and only if l°° is a subset. Thus l°° c ({x} —►F), or /°° -x c F. Thus [b] => [c]. Finally suppose

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[c]. Let Tx be the multiplier map from /°° to F defined by Tx(y) = x • y . By the Closed Graph Theorem, all multiplier maps between FAT-spaces are continuous [20]. Since MA is a bounded subset of /°°, TX(MA)= x -MA is a bounded subset of F . Thus x has the property \AB\ in F. □

Since e e l°° , we have (/°° —►F) c F. Anderson and Shields [1] have observed that (/°° —►F) is the largest solid subspace of F. By [c] above, we obtain the following. Corollary 1. Let E be an FK-space containing cp. Then E,AB, = (l°° —►F). This is the largest solid subspace of E. Corollary 2. An FK-space has the property \AB\ if and only if it is solid. Corollary 3. For any FK-space E, EUAB is solid. Proof. If F is an FF-space under the seminorms p , k = 1,2,3, ... , then EUAB is an FF-space under the seminorms supAe^ p (h • x). Since MA = MAm• Mfn = Mf • MA, it follows that MA-x is a bounded subset of F if and only if MA• x is a bounded subset of EUAB. This is true if and only if MA■ x is a bounded subset of EUAB . That is, EUAB = (EUAB)UAB = (EUAB)lAB¡, which is solid. The space EUAB can be characterized as follows. Theorem 3. Let E be an FK-space containing cp. Then EUAB= (c0 —►F). Proof. By [12, Theorem 4] cQ-EUABc F. Thus EUABc (c0 -» F). Conversely, suppose c0 • x c F. Define the map Fx : c0 —►F by I^GO = x -y. Since T is continuous, 7^ takes bounded subsets of c0 into bounded subsets of F . Let U be the unit sphere of c0 . Then U • x is bounded in F . By [12, Theorem 3], we have x € EUAB . Corollary. Let E be an FK-space containing cp. Then E n EUAB= (c —►E). In the same way, we can use the results in [9] to show that EAB = (bv0 -» F) and F n EAB = (bv -» F). Although F^^^ is solid, EUABn F need not be solid. This is the case when E = c. Thus, E,AB, is generally a proper subspace of EUAB n F. Also, if F c EUAB, the space F^^^ need not be the smallest solid space containing F . The space (c0)UAB= A provides an example. Theorem 4. Let E be an FK-space. If EUABc F, then E,AB, = EUAB. Proof. Clearly, E,AB, C EUAB. By Theorem 2, Corollary 3, EUAB is solid. Since E,AB, is the largest solid subspace of F, the statement follows. D

Theorem 5. Let E bean FK-space containing cp. Then the following statements are equivalent: [a] F ¿s solid and has the property AD ;

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[b] F is solid and has the property AK ; [c] E = c0-E; [d] E has the property UAK ; and [e] E has the property \AK\. Proof. The equivalence of [a], [b], and [c] was proved by Garling [9, p. 1007]. [b] => [e] : Fremlin and Garling [9, p. 1006] showed that a solid FF-space F is locally solid; that is, the topology of F is defined by seminorms p with the property p(d-x) < p(x) for all sequences d in the unit sphere of l°° . Let p be such a continuous seminorm, let x e E, and let e > 0. Then p,E, = p . Suppose p(snx - x) < § , and let h e M' such that h > s" . Then s" = h • s" . Hence P|£|(/z-x-x) < P\E\(h-x-snx)+P\E\(snx-x) = P\E\(h-x-h-snx)+p\E\(s"x-x) = p(hAx~ s"x)) +p(s"x - x) < lp(s"x - x) < e . This shows that the net MA• x converges to x under the topology of E,AB,. [e] =>■[d] is immediate from the definitions, [d] =4>[b] : The property UAK clearly implies AK, since s" e MA for all n . Sember and Raphael [13, Corollary 3.2] have shown that EUAK is solid. Since F = EUAK, it follows that F is solid. D Remark. A FF-space has the property \AK\ if and only if, for all x e E, \\s"x - x|||£, —>0 as «-»oo. Moreover, in this case ||x||,£| = sup ||i"x|||£,. A similar statement can be made about FF-spaces. Theorem 6. If E is an FK-space containing cp and EAD is solid, then EAD = âk - ûak = E\ak\ ~ co ' E ■ Proof. Clearly E,AK, c EUAK c EAK c EAD . Since EAD is a closed subspace of F, it is an FF-space under the subspace topology. Hence E,AK, = (EAD),AK,. By Theorem 5 ([a] => [e]), we have (EAD)]AK]= EAD . D Corollary 1. If E is an FK-space containing cp with the property UAB, then EAd = EAK = EUAK = Fî = c0- E. Proof. If F has the property UAB, then by Sember and Raphael [ 13, Theorem 4], EAD = EUAK = c0 • F. Thus EAD is solid and satisfies the conditions of Theorem 6. D If F is solid, then F has the property UAB since F = E,AB,c EUAB. Corollary 2. If E is a solid FK-space containing cp, then EAD = EAK = EUAK= F — r • F

Corollary 3. For any FK-space E containing cp, E,AK, = c0 • E,AB,. Proof. By definition, E,AK, = (E,ABAUAK. Since E,AB, is solid, we have

(E\ab\wak = co ' E\ab\ by Corollary 2. D Theorem 7. Let E be an FK-space. Then EUAK = E,AK,.

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Proof. Let x e E. The statement x e EUAK means that the net MA• x converges to x in the topology of F. The statement x e E,AK, means that the net MA■ x converges to x in the topology of E,AB, ; that is, E,AK, = (E,ABAUAK. Also EUAK = c0 ■EUAB = c0 ■(EUAB)UAB = (EUAB)UAK . It remains to be shown that (E,ABAUAK= (EUAB)UAK,which by Theorem 6, Corollary 2 is equivalent to (E¡ABl)AK = (EUAB)AK. Since E]AB] C EUAB, we have the inclusion (e\ab\)ak c (euab)ak- Conversely, suppose x e (EUAB)AK.Then, for each continuous seminorm p on F, supAe;r p(h ■(snx - smx)) —»0 as n, m -* oc . Since s"x - smx e cp, we have suph€jr p(h ■(snx - smx)) —>0 as n, m —►oo ; that is, x€ (FMß|)^. D

4. The solid hull of an FF-space For an FF-space F, the solid hull (F) is the intersection of all solid FF- spaces containing F. It is clearly solid. The solid hull was investigated by Anderson and Shields in [1]. We show that the solid hull is an FF-product space, and we find a dual relationship between E,AB, and (E). The FF-product E®F of two FF-spaces F and F was defined in [6] and [7] and was characterized as the smallest FF-space containing the coordinate product E • F. If F and F are FF-spaces, then EF turns out to be a FF-space. Theorem 8. Let E be an FK-space. The solid hull of E is the FK-space /°°®F. Proof. If F is a solid FF-space containing F, then F = /°° • F D /°° • F D F. Thus F D /°°®F d E. But 1°°®E is itself solid, since r®(l°°®E) = (/00(gi/00)(8)F= /°°®F. Thus it is the smallest solid FF-space containing F. D [7, Theorem 4.3] states that (E®F)AD = (E®F)AK = EAKFwhenever F c F^B . We obtain the following:

Corollary. Let E bean FK-space containing cp. Then (1°°®E)ADad =- v(/°°®F) ^^'AK c0®E. From this corollary we see that if F is solid and has the property AD, then E = c0- E. This is Theorem 5 ([a] => [c]). Other parts of Theorems 5 and 6 can also be obtained. The next theorem exhibits a dual relationship between the space E,AB, and the solid hull (E). Theorem 9. Let E and F be FK-spaces. Then ((E) -* F) = (E —>E),AB,. That is, the F-dual of the solid hull of E is the largest solid subspace of the F-dual of E. Proof. By Theorem 2, Corollary 1, (E -+ F)lAB{= (l°° -* (E -» F)). By [7, (5.6)], (A - (E - F)) = t(l°°êE) - F). D

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For example, let the a- and /Fduals of F be defined by Ea = (E -* ll) and Eß = (E -> cs), respectively. We have Ea = (Ea)]ABl = (E)a. Also Íe")\ab\ = (/0° - E") = (£ßß - *') = £ßßa = E° t7' Theorem (5.1)]; [8, Theorem 1]. Similarly ((E)/ = Ea . Corollary. For any FK-space E, we have Eaa = (E)AB = (E)UAB . Proof. By Theorem 9, we have (E)a = (Ea),AB,. Also, (Ea),AB, = Ea, since Ea is solid. Thus (E)aa = Eaa . But (E)aa = (E)AB by [5, Theorem 4] and [8, Remark (6)]. As noted after Theorem 3, (E)AB = (bv0 —>(F)). This is a subset of (c0 -> (E)), which is EUAB by Theorem 3. That is, (E)AB C (E)UAB , and thus (E)AB = (E)UAB . D

5. Examples and applications The properties \AB\ and \AK\ are strong properties of FF-spaces. We have the following list:

l\AK\ - C0' l\AB\ - " I — l > c\ak\ = c\ab\ = co ' (q = ' ;

CS\AK\ ~ CS\AB\ - ' ' (C5) = C0 ' %a:[ = bv\AB\= /!. {bv) = l°°; l\AK\=l\AB\=:{lP) = lP(lAB, is a FF-space under the norm || • || since lrlCjl > || • II > rkr| = sup^^, > supfc/7*Cr|. Furthermore ||x|| > EJaI for all x £ (Ct-)^^! • □ Every series-sequence regular matrix F satisfies the conditions of Theorem 10. This can be shown by considering the sequences e , k = 1, 2, 3, ... .

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Corollary. If T is a series-sequence regular matrix, then (cT),AB,—I . We now apply the concepts considered in this paper to the spaces of Fourier coefficients of some classes of functions. Let Lp (p > 1) be the Banach space of all real- or complex-valued In -periodic integrable functions with the norm 11/11i' = (¿F / I/O ' wnere the integral is taken over any interval of length In. Let C be the Banach space of all continuous real- or complex-valued 27T-periodic functions with the norm ||/||c = supx |/(x)|. If / e L , let f'k), k e Z, denote the kth complex Fourier coefficient of /. / = (f(k))k€Z and let snf, « = 0,1,... denote the «th partial sum of the Fourier series of /. If F is a subspace of F1 , let F denote the class of all sequences of Fourier coefficients of functions in F, i.e., F = {/: f e E} . Although the results in the preceding sections are for spaces of one-way sequences, they can be easily extended to the classes F of two-way sequences. If F is a Banach space, then F is a Banach space under the induced norm ll/llf := ll/llf > and conversely. Given a Banach space F contained in F we can determine the corresponding subspaces of absolutely bounded and absolutely convergent Fourier series, in the topology of F, by determining the spaces E\ABi and E,AK,. We shall also consider the corresponding solid hull (F). Two classical spaces of functions in Fourier analysis, determined by the pointwise convergence, ordinary / and absolute |/|, are the spaces of uniformly and absolutely convergent Fourier series: V = {feC:sJ-*fI uniformly} and A = {/ e C: sj - /|/| a.e.}. They are Banach spaces, under the norms: 11/11*:=sup \\snf\\c and ||/|L, := £|/(fc)| = U/H,,. kez It is well known that A C % C C c F°° properly, where L°° is the corresponding space of essentially bounded measurable functions. We shall also consider the Banach space M of In -periodic Radon measures, under the norm II/IU= supj¿t£Lo^/IIf- For the spaces F = Lp(p > 1) and F°° , the questions of determining the largest solid space contained in F and the smallest solid space containing F have already been considered in [1]. Slightly expanding those results in view of the concepts of this paper, we can write the following theorem, where the standard sequence spaces are to be interpreted as the spaces of two-way sequences: Theorem 11 [a] If E is a Banach space and L c F c F , then E,AK,= E\AB\= / and I2 = (L2) c (F) c (V) = c0. Moreover, ifll and l/p +l/q=l, then A2 c Lp and (Lp) = I2.

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[c] If E is a Banach space and A c F c F°°, then E,AK, = E,AB, = /' and /' = (A) c (Ê) C (L°°) C I2. [d] MlAB]=LllABl=l2 and (M) = T.

[a] It was pointed out in [1] that L,AB, = I = L,AB]. Thus, E,AB, = / and, since / 2 has AD, by Theorem 5 we have that E,AK, = E,AB,. The corresponding statements about solid hulls were discussed in [1], and the last statement is a corollary of a result in [11]. [b] The inclusion follows from [11], and the equality (Lp) = I2 forp>l was also discussed in [1].

[c] Since A = /' is solid, s/,AB,r*\AB\ -'= ll ■. The**»«M™"«J equality ^\ABLA, = /' was explained in [1]. Thus E\AB\= / and, by Theorem 5, E\AK, = E,AB,■ The inclusion about solid hulls is obvious. [d] Clearly L1^ c M]AB] and, by [a] L^ = I2. Hence I2 C A/|/1Ä|. Conversely, let / 6 M{AB¡. Then ||/|||i?| < oo ; i.e., sup„ ||^A E*-0**/H|£»| < oo . Since L,¿B, = / , we have sup„ ||^A X)i=oJ/t/ll/2 < °° and therefore 2A/2 isupf £ |/(*)|2|

Thus f el . This proves the first inequality. To show that (A/) = /°° , we first note th£U e e M, so that e-l°° = l°° c (M). But A? c /°° implies (A?) c l°° . Thus (M) =r. O Corollary. s&\AB\= &\Ab\ = ^\ab\ ~ e<\ab\ = ' = A, a«ú? //lé-same sín'«g o/ equalities is true for \AK\. We consider now some newer classes of functions introduced in Fourier analysis. They are determined by other types of pointwise convergence, namely strong convergence of index p > 1 , [/] , and absolute convergence of index p > 1, |/| . The latter extends the concept of absolute convergence IF in the sense that a sequence sn -> j|/| if and only if sn —>si and J2kp~ \sk - sk_x\p < oc. The strong convergence [/] lies between the absolute |/| and the ordinary convergence /: that is, |/| =$■[I]p => I; see [14] or [16] and the references cited there. These notions were applied to trigonometric and Fourier series in a series of recent papers, [16] through [18], which led to the study of the related spaces of functions, [14], [15], [19]: Sp = {feLx: sj -» f[I]p a.e.}, Sp = {/ e C: sj - f[I]p uniformly}, Ap = {feLl: sj - /l/l, a.e.}, Ap = {/ e C: sj - f\I\p uniformly}. For p = 1, we write simply S, S, A, and A , respectively. They have many interesting properties: Sp c Sp c f)\

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the classes Sp and S" decrease with p increasing while the classes s/p are incomparable and the same is true for Ap ; A = A ; sfp c §p C %Aand Ap c Sp c Lp properly. From the results in [14], [15], and [17] they can be described as follows. For p > 1, let sp := {x: ^ Ew oo)}. Then _ cP — fy, „P-l V S = V ni1 and S c s1 properly, S" = sp = {x: aA1 A,|>n'|/t|>n l^fc|x/ = o(l) (« -* oo)} for p > 1, and SP = C n S" for p > 1. For p > 1, lp = *; \\f\y = 11/11*+ ll/llw for p > 1; |^, = ||/I|W and 11/11^= 11/11*+ ||/|||p| for p > 1, where i/p w = su^2«irT^(|/:| + 1)/'l/w|/'l and \k\

ipa| i/(o)r+ x i^r'i/wi k&Z.k^tO

Theorem 12 [a] S\ak\ = S\ab\ =s"=Sp = (Sp) for p>\. [h] S\ak\ = S\ab\ = I n s (* and S 1 a/i«/ (§) c/2nA 5|ylfi|.

Proo/ [a] By the above remarks, Sp = sp for p > 1 . Since sp is solid and has the property AD, the statement follows from Theorem 5. [b] S = L m1 and, consequently, S,AB, C £A, n$', since sl is solid. A1 2 *"* 2 1 By Theorem 11 [a], L,AB>= I , and therefore S,AB, c / flj , Conversely, / n s c L|^B| n s c $iAb\ ■ Moreover, by Theorem 5, S,AK,= S,AB,. [c] By the above remarks, Sp = C n Sp and, by the corollary of Theorem 11, ¿A, = / . Hence, by statement [a], S,AB, c /'n/ and conversely / Dsp c C,AB,nSjA c cA5,. The equality S,^, = S,^, is clear from Theorem 5. [d] Since Sp = Cf)Sp , clearly (SP) C (Sp) = s" for all p > 1 . To show the converse inclusion for p > 1 , we refer to a result due to Salem [2, vol. 1, p.

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335], noting first that x 6 / implies that

hk^^\kky^ —?— y^ w21 Since sr° c s¿ for p > 2, it clearly suffices to assume that 1 < p < 2. Taking 1 < p < 2 for x e sp we have, 1/2 / N 1/P E w2 ) ^ E i*/ - ° (¿tí) - wherei/P+1/« -1,

from which it follows that (*) is satisfied. We now show that sp c l°° • Sp . For x e sp, let x = xr + x , where xk = xk for k > 0, x[ = 0 for k < 0, Il r and xk = 0 for k > 0, xk = xk for k < 0. By the above argument, both x and x satisfy (*). Consequently, by Salem's theorem, there exists a sequence (a¿)^=o such that the series oo X x¿cos(/cí - û£) fe=0 converges uniformly and is therefore the Fourier series of its sum function g e C. Hence gc(k) = xk cosak and gs(k) = xk sinak . Expressed in complex form, g is the uniform sum of the series J2 S(k)e' ', where g(k) = \x[e~lak, g(-k) = \x[e'ak. iez Consequently, defining a two-way sequence y by yk = le'ak for k > 0 and yk = 0 for k < 0, we have xr = y • g where y e l°° . But clearly g e sp = Sp , and therefore xr = y-g e l°°-Sp . In the same way we can show that x1 e l00-^ . Consequently x 6 1°°®$" c (Sp). The corresponding properties for the spaces Ap and A p are proved similarly, noting that ap c sp so that x e ap implies (*). D Theorem 13 [a] ¿\ak\ = ¿\ab\ =ap = Ap = (Ap) for p > 1. [b] ^1=^1 = /'no"/or p>l. [c] (A p) = ap = Ap for p > 1 anrf (A) = /'.

References

1. J. M. Anderson and A. L. Shields, Coefficient multipliers of Bloch functions, Trans. Amer. Math. Soc. 224 (1976), 255-265. 2. N. Bary, A treatise on trigonometric series, vols. 1 and 2, Pergamon Press, New York, 1964. 3. G. Bennett and N. J. Kalton, Inclusion theorems for K-spaces, Cañad. J. Math. 25 (1973), 511-524.

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4. M. Buntinas, Convergent and bounded Cesàro sections in F K-spaces, Math. Z. 121 (1971), 191-200. 5. _, On Toeplitz sections in sequence spaces, Math. Proc. Cambridge Philos. Soc. 78 (1975), 451-460. 6. _, Products of sequence spaces, Analysis 7 (1987), 293-304. 7. M. Buntinas and G. Goes, Products of sequence spaces and multipliers, Radovi Mat. 3 (1987), 287-300. 8. D. J. H. Garling, The ß- and y-duality of sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967), 963-981. 9. _, On topological sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967), 997-1019. 10. C. Goffman and G. Pedrick, A first course in functional analysis, Prentice-Hall, Englewood Cliffs, NJ, 1965. 11. C. N. Kellogg, An extension of the Hausdorfif-YoungTheorem, Michigan Math. J. 18 (1971), 121-127. 12. J. J. Sember, On unconditional section boundedness in sequence spaces, Rocky Mountain J. Math. 7(1977), 699-706. 13. J. Sember and M. Raphael, The unrestricted section properties of sequences, Canad. J. Math. 31 (1979), 331-336. 14. I. Szalay and N. Tanovic-Miller, On Banach spaces of absolutely and strongly convergent Fourier series, Acta Math. Hung. (1989), (to appear). 15. _, On Banach spaces of absolutely and strongly convergent Fourier series, II, Acta Math. Hung, (to appear). 16. N. Tanovic-Miller, On strong convergence of trigonometric and Fourier series, Acta Math. Hung. 42(1983), 35-43. 17. _, On a paper ofBojanic and Stanojevic, Rendiconti Cir. Mat. Palermo 34 ( 1985), 310— 324. 18. _, Strongly convergent trigonometric series as Fourier series, Acta Math. Hung. 47, (1986), 127-135. 19. _, On Banach spaces of strongly convergent trigonometric series, J. Math. Anal, and Appl. (to appear). 20. K. Zeller, Allgemeine Eigenschaften von Limitierungsverfahren, Math. Z. 53 (1951), 463- 487.

Department of Mathematical Sciences, Loyola University of Chicago, Chicago, Illi- nois 60626 Department of Mathematics, University of Sarajevo, 71000 Sarajevo, Yugoslavia

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