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Sequence Spaces with Oscillating Properties
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
200, 519]537 Ž1996.
0222
Sequence Spaces with Oscillating Properties Johann BoosU Fachbereich Mathematik, Fernuni¨ ersitat}Gesamthochschule, Postfach 940, D-58084 ¨ Hagen, Germany
Daniel J. Fleming† Department of Mathematics, St. Lawrence Uni¨ ersity, Canton, New York 13617
and Toivo Leiger ‡ ¨ Puhta Matemaatika Instituut, Tartu Ulikool, EE 2400 Tartu, Estonia Submitted by Brian S. Thomson Received February 18, 1994 DEDICATED TO PROFESSOR K. ZELLER ON THE OCCASION OF HIS
70TH BIRTHDAY In this note we consider various types of oscillating properties for a sequence space E being motivated by an oscillating property introduced by Snyder and by recent papers dealing with theorems of Mazur]Orlicz type and gliding hump properties. Our main tools, two summability theorems, allow us to identify two such oscillating properties for a sequence space E one of which provides a sufficient condition for E ; F to imply E ; WF while the other affords a sufficient condition for E ; F to imply E ; SF . Here F is any Lw -space, a class of spaces which includes the class of separable FK-spaces, SF denotes the elements of F having sectional convergence, and WF denotes the elements of F having weak sectional convergence. This, in turn, is applied to yield improvements on some other theorems of Mazur]Orlicz type and to obtain a general consistency theorem. Furthermore, combining the above observations with the work of Bennett and Kalton we obtain the first oscillating property on a sequence space E as a * E-mail: [email protected]. † E-mail: [email protected]. ‡ E-mail: [email protected]. 519 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
520
BOOS, FLEMING, AND LEIGER
sufficient condition for E b, the b-dual of E, to be s Ž E b, E . sequentially complete whereas the second assures both the weak sequential completeness of E b and the AK-property for E with the Mackey topology of the dual pair Ž E, E b .. Q 1996 Academic Press, Inc.
1. INTRODUCTION Numerous authors have used various gliding hump properties imposed on the multiplier space of a sequence space E which are weaker than the assumption that E be solid but are nonetheless sufficient to obtain consistency theorems or, equivalently, to establish the weak sequential completeness of E b, the b-dual of E, and thus getting theorems of Mazur]Orlicz type Žsee w2x for references .. Recent experience Žsee e.g., w8, 12, 11, 1x. has shown that it is sufficient to look for gliding hump properties satisfied for each x in E, that is, ‘‘pointwise’’ gliding hump properties, rather than imposing a uniform hump property on E via the multiplier space M Ž E .. For example, Noll in w8x introduced the so-called weak gliding hump property and showed that it is a sufficient condition for the sequential completeness of the b-dual. Stuart w11x extended Noll’s result for the signed weak gliding hump property. The latter property captures the space bs whereas the former does not. See also w6x for additional results on the signed weak gliding hump property. Furthermore, in w1, 11x Žsee also w6x. it is pointed out that the weak gliding hump property and, more generally, the signed weak gliding hump property of a sequence space E implies not only the weak sequential completeness of E b but also the AK-property of the Mackey topology of the dual pair Ž E, E b .. This proves that there is a big gap between the class of sequence spaces having the signed weak gliding hump property and the class of sequence spaces with weakly sequentially complete b-duals. In earlier papers of the first and third authors it is shown that X [ Y l WE has weakly sequentially complete b-dual when the sequence space Y has a suitable gliding hump property and E is an FK-space containing the finitely non-zero sequences. Here WE denotes the set of all elements of E which are the weak limits of their sections. Although such spaces X have weakly sequentially complete b-duals they will, in general, fail both the signed weak gliding hump property and the AK-property with respect to the Mackey topology of the dual pair Ž X, X b .. Therefore it is of mathematical interest to identify a class of sequence spaces with weakly sequentially complete b-duals which contains both the class of sequence spaces with the signed weak gliding hump property and the class of spaces Y l WE described above. This goal leads us naturally to the class of sequence spaces having so-called oscillating properties.
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
521
Our main tools are Theorems 3.2 and 3.3, which should be of independent interest to summability theorists since they assert that if a matrix B sums a sequence x having suitable properties then it cannot sum a lot of sequences of the type yx, where y is a suitable oscillating sequence. As a consequence we obtain several theorems of Mazur]Orlicz type which improve upon theorems in w2, 6, 1x.
2. NOTATION AND PRELIMINARIES Let v denote the linear space of all scalar Žreal or complex. sequences. By a sequence space E we shall mean any linear subspace of v . A sequence space E with a locally convex topology t is called a K-space if the inclusion map i: Ž E, t . ª v is continuous, where v has the topology of coordinatewise convergence. A K-space with a Frechet topology is ´ called an FK-space. If, in addition, the topology is normable then it is called a BK-space. We will assume throughout familarity with the standard FK-spaces and their natural topologies as well as the properties enjoyed by these spaces Žsee, e.g., w13, 7x.. We refer the reader to w4x for a discussion of Lw-spaces, a class of K-spaces which includes, as a proper subset, the class of separable FK-spaces. For a sequence space E the multiplier space of E and the b-dual of E are given by M Ž E . s x g v < xy g E for each y g E 4 and Eb s x g v
½
Ý x k yk converges for each k
5
ygE ,
where xy denotes the coordinatewise product. Let e denote the sequence of ones and let e k s Ž d jk .`js1 be the kth coordinate vector. For x g v , n g N the nth section of x is n
Pn Ž x . s
Ý
xk ek.
ks1
If Ž E, F . is a dual pair then s Ž E, F ., t Ž E, F . denotes the weak topology and the Mackey topology, respectively. For a sequence space E and a linear subspace F of E b, Ž E, F . is a dual pair under the natural bilinear form ² x, y : s
Ý x k yk . k
522
BOOS, FLEMING, AND LEIGER
If E is a K-space containing w , the space of finitely non-zero sequences, we let WE s x g E < Pn Ž x . ª x Ž s Ž E, EX . . 4 SE s x g E < Pn Ž x . ª x in E 4 , where EX denotes the topological dual of E. A K-space E containing w with E s SE is called an AK-space. If A s Ž a n k . is an infinite matrix with scalar entries the convergence domain
½
c A s x g v < Ax s
`
ž
Ý an k x k k
/
gc ns 1
5
admits a natural FK-topology w13x. For x g c A we write lim A x s lim Ax. If w ; c A let a k s lim n a n k and define
½
IA s x g c A
Ý ak x k exists k
5
,
L A : IA ª K by L AŽ x . s lim A x y Ý k a k x k Žwhere K s C or K s R., and LHA s x g IA < L A Ž x . s 0 4 . Further if w ; c A we write WA , SA instead of Wc A , Sc A . Obviously WA ; LHA . A sequence space E is said to be solid if m ; M Ž E . and monotone if m 0 ; M Ž E ., where m 0 denotes the linear span of all sequences of zeros and ones. In w8x Noll Žsee also w12x. introduced the weak gliding hump property of a sequence space E and showed that if E enjoys this property then E b is s Ž E b, E . sequentially complete. However, the space bs, which is known to have weakly sequentially complete b-dual, fails the weak gliding hump property. Motivated by this observation Stuart w11x extended Noll’s result to spaces which have the signed weak gliding hump property and he showed that bs satisfies the latter property if the scalar field is K s R. Here we introduce several new signed properties with the aim of obtaining improved theorems of Mazur]Orlicz type. A sequence Ž y Ž j. . in v is called a block sequence if there exists an index sequence Žg j . with g 1 s 1 and y kŽ j. s 0 for k - g j and g jq1 F k Ž j g N.. jq 1 y1 y Ž j. e k and the Therefore, each y Ž j. has the representation y Ž j. s Ýgks gj k Ž j. coordinatewise sum y s Ý j y is the limit of this series in Ž v , tv . and is given by y k s y kŽ j. for g j F k - g jq1 and j g N. By definition, a block sequence Ž y Ž j. . is a 1-block sequence if y kŽ j. s 1 Žg j F k - g jq1 and j g N..
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
523
A sequence Ž y Ž j. . in v is called a step 1-block sequence with respect to an index sequence Ž k i . with k 1 s 1 if there exists an increasing sequence Ž i j . in N such that:
Ža.
i 3 jy1 - i 3 j
Ž b.
y kŽ j. s
Žg .
½
Ž j g N. ,
0
if k - k i 3 jy 2 or k G k i 3 jq 1
1
if k i 3 jy 1 F k - k i 3 j
Ž ykŽ j. . is monotonic for k i
3 jy 2
Ž j g N . and y 1Ž1. G 0,
y 1 F k F k i 3 jy 1
and for k i 3 j y 1 F k F k i 3 jq 1
Žd.
ys
Ž j g N. ,
Ý y Ž j. Ž pointwise sum. is constant for k i F k - k iq1 Ž i g N. . j
Furthermore, it is called a step 1-block sequence if there exists an index sequence Ž k i . such that Ž y Ž j. . is a step 1-block sequence with respect to Ž k i .. In particular, each 1-block sequence is a step 1-block sequence and any step 1-block sequence Ž y Ž j. . fulfills 0 F y kŽ j. F 1 Ž j g N. and sup j 5 y Ž j. 5 b ¨ s 2. A 1-block sequence is said to be a 1-block sequence with respect to an index sequence Ž k i . if it is a step 1-block sequence with respect to Ž k i .. The definition of a step 1-block sequence is illustrated by the following picture.
In the following definitions the convergence of the sums is coordinatewise. DEFINITION 2.1. A sequence space E containing w has the signed pointwise gliding hump property ŽSIGNED P GHP. if for each x g E and every block sequence Ž y Ž j. . with sup j 5 y Ž j. 5 b ¨ - ` there exist a subsequence Ž y Ž n k . . of Ž y Ž j. . and a sequence Ž h k . in 1, y14 such that yx g E, where y [ Ý k h k y Ž n k ..
524
BOOS, FLEMING, AND LEIGER
Now other gliding hump properties can be defined depending upon additional properties enjoyed by the subsequence Ž y Ž n k . . in Definition 2.1. Ž2.1.1. If Ž y Ž n k . . can be chosen such that for each subsequence Žy of it there exists a sequence Ž h k . in 1, y14 with yx g E, where y [ Ý k h k y Ž m k . then we say E has the signed strong pointwise gliding hump property ŽSIGNED SP GHP.. Ž2.1.2. If Ž y Ž n k . . can be chosen such that for each subsequence Ž m Ž y k . . of it and each Ž h k . in 1, y14 we get yx g E, where y [ Ý k h k y Ž m k . then we say E has the absolute strong pointwise gliding hump property ŽABSOLUTE SP GHP.. Ž mk..
For convenience we use in the following the notion of a strong subsequence Ž y Ž jk . . of a sequence Ž y Ž j. . in the sense that N R jk < k g N4 is an infinite subset of N. DEFINITION 2.2. Let E be a sequence space containing w . E is defined to have the signed pointwise oscillating property ŽSIGNED P OSCP. if for each x g E and any index sequence Ž k i . with k 1 s 1 there exists a strong subsequence Ž y Ž n j . . of a step 1-block sequence Ž y Ž n . . with respect to Ž k i . such that there exists a sequence Ž h j . in 1, y14 with yx g E, where y [ Ý j h j y Ž n j .. Now similarly to the case of gliding hump properties other oscillating properties can be defined; in the following we need Ž2.2.1. If in Definition 2.2, Ž y Ž n . . can subsequence Ž y Ž n j . . of it and each sequence where y [ Ý j h j y Ž n j ., then we say E has oscillating property ŽABSOLUTE SP OSCP..
be chosen such that for each Ž h j . in y1, 14 we get yx g E, the absolute strong pointwise
We remark that the just defined oscillating properties are motivated by an oscillating property introduced by Snyder w10x and some recent papers connected with theorems of Mazur]Orlicz type and gliding hump properties. DEFINITION 2.3. We say E has the signed pointwise weak gliding hump property ŽSIGNED P WGHP. and the signed pointwise 01-oscillating property ŽSIGNED P 01-OSCP. if the definition of the SIGNED P GHP and the SIGNED Ž j. P OSCP is fulfilled for subsequences of 1-block sequences Ž y ., respectively. Note that the
SIGNED P
WGHP
is due to C. Stuart w11x.
Remark 2.4. Let E and Y be sequence spaces containing w . Ža. Žb. P
ABSOLUTE SP
GHP ; ABSOLUTE SP
ABSOLUTE SP
GHP ; SIGNED SP
WGHP ; SIGNED P
01-OSCP ; SIGNED
OSCP
; SIGNED
GHP ; SIGNED P P
OSCP.
P
OSCP.
GHP ; SIGNED
525
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
Žc. If Y has the SIGNED then E l Y has the SIGNED P
P
GHP
and E has the
ABSOLUTE SP
OSCP
OSCP.
If we replace in the above definitions of the gliding hump and oscillating properties Žsee 2.1, 2.2, and 2.3. the set 1, y14 by S [ z g K < < z < s 14 , where K denotes the scalar field, that is, we take in the case of complex sequences a signum sequence on the unit circle instead of a sign sequence, then we get more general definitions and all results of the paper remain true. 3. MAIN RESULTS First of all we identify large classes of sequence spaces having the GHP and the ABSOLUTE SP OSCP, respectively.
ABSOLUTE SP
THEOREM 3.1. ABSOLUTE SP
GHP
Let E be an FK-space containing w . Then S E has the whereas WE has the ABSOLUTE SP OSCP.
Proof. The first part of the statement can be proved by a refinement of the proof of w1, Theorem 3.3x. For a proof of the second, let x g WE and an index sequence Ž k i . be fixed. Furthermore, let ! ! be a paranorm generating the FK-topology t of E. According to w2, Lemma 1x Žsee also the text following w3, Remark 2.3x. we may choose a sequence Ž x Ž r . . in the convex hull of Pk iy1Ž x . < i g N4 such that in Ž E, t . ,
xŽr. ª x
Ž 1.
tr
xŽr. s
Ý mr i Pk y1Ž x . i
iss r
ž
tr
sr , t r g N with sr F t r - srq1 , 0 F m r i F 1, m r t r / 0,
Ý mr i s 1 iss r
/
. Ž 2.
By virtue of Ž1. we may choose an index sequence Ž r j . such that ! x Ž r . y x Ž rq m . !- 2yjy1
Ž m g N and
r G rj . .
Ž 3.
Then we define z Ž j. [ x Ž r 2 j . y x Ž r 2 jy 1 . Ž j g N .
z[
and
Ý z Ž j. Ž pointwise sum. . j
Because of the representation of x with z Ž j. s y Ž j. x, y kŽ j. s
¡0 ¢1
~
Žr.
in Ž2. we may choose y Ž j. Ž j g N.
if k - k s r if k t r
2 jy 1
2 jy 1
or k G k t r
F k - k sr . 2j
2j
526
BOOS, FLEMING, AND LEIGER
Furthermore, as we may check, Ž y Ž j. . is a step 1-block sequence with respect to Ž k i .. From Ž3. we get that for each subsequence Ž y Ž k j . . of Ž y Ž j. . and each Ž h j . in S the sequence ŽÝ Njs1 h j y Ž k j . x .N is a Cauchy sequence in Ž E, t .. Therefore, because of the completeness of Ž E, t . and the coordinatewise convergence, we have
˜yx g E
with ˜ y[
Ý h j y Ž k . Ž coordinatewise sum. . j
j
Evidently, ˜ y and Ž h j y Ž k j . . satisfy the conditions in Lemma 2 in w2x, thus ˜yx g WE . That completes the proof of the theorem. Note that we have considered in the above proof more general sequences Ž h j . in S . The following ‘‘non-summability theorems’’ form the basis for the main results of this paper and should be of independent mathematical interest. THEOREM 3.2. Let B be a matrix with w ; c B and let x g c B be gi¨ en such that at least one of the following statements Ži., Žii. is fulfilled: ny1 b x / Ži. There exists an index sequence Žhn . such that lim n Ýhks1 k k lim B x. Žii. supn <Ýnks1 bk x k < s `.
Then there exists an index sequence Ž k i . such that for each strong subsequence Ž y Ž m j . . of any step 1-block sequence Ž y Ž m . . with respect to Ž k i . we ha¨ e z [ yx f c B , where y [ Ý j h j y Ž m j . Ž pointwise sum. and Ž h j . is any sequence in S . Proof. First of all we make some considerations in advance. In both case Ži. and case Žii. we will choose an index sequence Ž k i . depending upon x g c B and information resulting from this fact and w ; c B . For all index sequences Ž k i . and Ž n i . and any sequence z g v we use the notations
Ý bn k z k s A i q AUi q Bi q Ci i
Ž i g N. ,
k
where the convergence of Ý k bn i k z k is assumed, k iy1
Ai [
Ý Ž bn k y bk . z k , i
AUi [
ks1
Ý ksk i
Ý
bk z k ,
ks1
k iq1 y1
Bi [
k iy1
bn i k z k
and
Ci [
`
Ý ksk iq1
bn i k z k .
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
527
In all cases we will construct the index sequences Ž k i . and Ž n i . such that
Ž A i . g c0
and
Ž Ci . g c0 ,
where z is chosen as is described in the theorem. It is then immediate that each of the following conditions implies z f c B : Ž a . Ž AUi . g c and Ž Bi . f c. Ž b . Ž AUi . f c and Ž Bi . g c 0 , where Ž i j . is a suitable index sequence. j j Let x g c B and let Žhn . with h1 s 1 be any given index sequence. ŽLater on we will fix Žhn . on the base of the properties Ži. and Žii... Since w ; c B and x g c B we may choose index sequences Ž n i ., Ž k i ., and Ž n i . having certain properties: For n 1 [ 1 and k 1 [ hn 1 we may choose an n1 g N such that k 1y1
Ý
< bn k y bk < < x k < - 2y1
Ž n G n1 . .
ks1
Then we may choose n 2 ) n 1 such that for k 2 [ hn 2 L
Ý bn k x k
- 2y2
Ž n F n1 , k 2 F l - L . .
ksl
If we have chosen n iy1 and n i then for k i [ hn i we determine n i ) n iy1 with k iy1
Ý
< bn k y bk < < x k < F 2yi
Ž n G ni . ;
Ž 4.
ks1
furthermore, we choose n iq1 ) n i such that for k iq1 [ hn iq 1 we have L
Ý bn k x k
- 2yŽ iq1.
Ž n F n i , k iq1 F l - L . .
Ž 5.
ksl
Now, let Ž y Ž m . . be any step 1-block sequence with respect to Ž k i .. Then we consider in both cases strong subsequences Ž y Ž m j . . of Ž y Ž m . .. Using 5 y 5 ` F 1 for y [ Ý j h j y Ž m j ., where h j g S Ž j g N., and noting Ž d . in the definition of a step 1-block sequence we get for z [ yx the estimations k iy1
Ý ks1
< bn k y bk < < x k < i
iª`
0
6
< Ai < F
528
BOOS, FLEMING, AND LEIGER
because of Ž4. and `
`
k rq1 y1
Ý
Ý
rsiq1
ksk r
bn i k z k F
Ý ksk iq1
`
bn i k x k F
2yr
Ý
iª`
0
6
< Ci < F
rsiq1
because of Ž5.. Now, we shall fix Žhn . depending upon Ži. and Žii.. In case Ži. we may choose Žhn . such that h1 s 1 and
a [ lim n
hny1
Ý
bk x k / lim B x \ d;
ks1
furthermore we may assume hnq m y1
bk x k - 2yn
Ý
ks hn
Ž n , m g N. .
Ž 6.
For this Žhn . let Ž k i ., Ž n i ., and Ž n i . be chosen as is described above. Furthermore, let Ž y Ž m . . be any step 1-block sequence with respect to Ž k i . and let Ž y Ž m j . . be any strong subsequence of Ž y Ž n . .. Since it is strong there exists an index sequence Ž jr . with m j r q 1 / m j rq1 for each r g N. Roughly speaking, if we consider y [ Ý j y Ž m j . Žcoordinatewise sum. then there exists a 0-block in between y Ž m jr . and y Ž m jrq 1 . ; in particular, if Ž i j . is chosen as in the definition of a step 1-block sequence, then we have yk s
¡0 ¢1
~
if k - k i 3 m
or k i 3 m
1y 2
if k i 3 m y 1 F k - k i 3 m j
j rq 1
F k - k i3m
2 j rq 1 y
Ž r g N. Ž 7.
Ž j g N.
j
as we may verify with Ž b . in the definition of a 1-block sequence by the aid of the picture following that definition. We will prove z [ yx f c B , where y [ Ý j h j y Ž m j . and h j g S Ž j g N.. Let r , l g N with 1 F l - r be given. By that and Ž6. we get kry1
Ý ks1
bk z k y
kl y1
Ý
k r y1
bk z k s
ks1
bk z k
Ý kskl
nry1 hm q1 y1
Ý
ms nl
Ý
ks hm
bk x k F
implying the existence of k iy1
lim i
Ý ks1
bk z k
Ý
m s nl
2ym
lª`
0,
6
F
nr y1
529
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
and thus Ž AUi . g c. Now, we prove Ž Bi . f c. By construction we know that zk s
¡h x ¢0
~
j
if k i 3 m y 1 F k - k i 3 m
k
j
if k i 3 m
F k - k i3m
j rq 1
Ž j g N.
j
Ž r g N. .
j rq 2
Thus Bi 3 m s 0 by Ž7.; therefore Ž Bi 3 m . g c 0 . Then Ž Bi . f c is proved j rq 1 j rq 1 if Ž Bi 3 m . f c 0 . However, if one puts i [ i 3 m j y1 , this follows from the j ry 1
r
identities k iq1 y1
Bi s
k iq1 y1
bn i k z k s h j r
Ý ksk i
s h jr
ž
bn i k x k
Ý ksk i
k iy1
Ý bn k x k y Ý i
k
k iy1
/
bk x k y h jr
ks1
`
Ý Ž bn k y bk . x k y h j Ý i
r
ks1
bn i k x k
ksk iq1
iy1 b x . because of Ž4. and Ž5. and the fact that ŽÝ`ks 1 bn i k x k y Ý kks1 k k igN U converges to d y a / 0. Altogether, we proved Ž A i . g c and Ž Bi . f c, thus z f c B by Ž a .. In case Žii., without loss of generality, we may assume
sup R n
ž
n
Ý bk x k ks1
/
s `.
Therefore, we may choose an index sequence Žhn . such that h1 s 1 and R
ž
hnq1 y1
Ý
ks hn
hn y1
/
bk x k G n q
Ý
< bk x k <
Ž n g N. .
Ž 8.
ks1
Again, for this Žhn . let Ž k i ., Ž n i ., and Ž n i . be chosen as is described above. Furthermore, let Ž y Ž m . . be any step 1-block sequence with respect to Ž k i ., let Ž y Ž m j . . be any strong subsequence of Ž y Ž m . ., and let Ž y Ž m jr . . be chosen as in case Ži.. We will prove z [ yx f c B , where y [ Ý j h j y Ž m j . and h j g S Ž j g N.. Considering i [ i 3 m j q1 we get by Ž7. and the definition of z the r statement Bi s 0, thus Ž Bi . g c 0 , and also Ž AUi . f c since <
AUi <
k iy1
s
Ý
k i 3m
bk z k G < h j r <
ks1
y1
k i 3m y1 jr
ksk i 3m
0
bk x k y
Ý j ry 1
bk x k y kŽ m j r . y
Ý ksk i 3m
k i3 m
j ry 1
y1
bk z k
Ý ks1
j ry 1
k i3 m
j ry 1
y1
Ý ks1
< bk z k < G 3 m j
rª` r
`,
6
GR
j rq 1
530
BOOS, FLEMING, AND LEIGER
where we made use of R
ž
hnq1 y1
Ý
ks hn
/
Ž n g N. ,
bk x k G 0
which follows from Ž8.. Altogether we have shown z f c B by Ž b . and statement Ža. is proved. THEOREM 3.3. Let B be a matrix with w ; c B and let x g c B R SB be gi¨ en. Then there exists an index sequence Ž ln . such that for each strong subsequence Ž y Ž m j . . of any 1-block sequence Ž y Ž m . . with respect to Ž ln . we ha¨ e z [ yx f c B , where y [ Ý j h j y Ž m j . Ž pointwise sum. and Ž h j . is any sequence in S . Proof. We assume x g LHB RSB Žotherwise we may apply Theorem 3.2.. First of all we remark that x f SB if and only if there exists an « ) 0 such that for each n g N there exist index sequences Ž nn j . j and Ž bn j . j with bn jy1 < bn 1 ) n such that <Ý ks n bnn j k x k G « . Let « ) 0 and let the index sequences be chosen in that way. We put k 1 [ 1 and choose n1 , k 2 g N with k 2 ) k 1 such that 1
Ý < bn k y bk < < x k < - 2y1
Ž n G n1 .
ks1
and k 2y1
bn1 k x k G « .
Ý ksk 1
After that we may choose n 2 , k 3 g N with k 3 ) k 2 and n 2 ) n1 such that k 2y1
Ý
< bn k y bk < < x k < - 2y2
Ž n G n2 . ,
ks1 k 3y1
Ý
bn 2 k x k G «
ksk 2
and L
Ý bn k x k ksl
- 2y2
L
and
Ý bk x k ksl
- 2y2
Ž n F n1 , k 3 F l F L . .
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
531
Having fixed n1 , . . . , n i and k 2 , . . . , k iq1 we choose n iq1 , k iq2 g N with k iq2 ) k iq1 and n iq1 ) n i such that k iq1 y1
Ý
< bn k y bk < < x k < - 2yŽ iq1.
Ž n G n iq1 . ,
ks1 k iq2 y1
bn iq 1 k x k G «
Ý
Ž 9.
ksk iq1
and L
L
- 2yŽ iq1.
Ý bn k x k
- 2yŽ iq1.
Ý bk x k
and
ksl
ksl
Ž n F n i , k iq2 F l F L . . Now, we put ln [ k 2 ny1 Ž n g N. and let Ž y Ž m . . be a 1-block sequence with respect to Ž ln . and Ž y Ž m j . . be a strong subsequence of Ž y Ž m . .. Since it is a strong subsequence there exists an index sequence Ž jr . with m j r q 1 / m j rq1 Ž r g N.. For a representation of y Ž m . Ž m g N. we may choose an index sequence Ž nm . with y kŽ m . s
½
1
if lnm F k - lnm q 1
0
otherwise
Ž m , k g N. .
Thus, for y [ Ý j h j y Ž m j . Žpointwise sum., where Ž h j . is any sequence in S , we have yk s
¡h ¢0
~
if lnm s k 2 nm y1 F k - lnm q 1 s k 2 nm q 1y1
j
j
j
if k - lnm or lnm 1
j
j rq 1
j
F k - lnm
j rq 1
.
Ž 10 .
Using the notations for A i and AUi as in the proof of 3.2 and k iq2 y1
Bi [
bn i k z k
Ý
and
Ci [
ksk i
`
bn i k z k
Ý ksk iq2
we get 0,
Ž AUi . g c,
Ci
iª`
0
6
iª`
6
Ai
for z [ yx quite similarly to the corresponding cases in the proof of 3.2.
532
BOOS, FLEMING, AND LEIGER
Thus, z f c B is proved if Ž Bi . f c. But this follows by the following considerations: If r g N and i [ 2 nm j q1 y 2 then by Ž9. and Ž10. r
< Bi < s
k iq1 y1
Ý
bn i k z k s < h j <
ksk i
k iq1 y1
Ý
bn i k x k G « ) 0
ksk i
and if i [ 2 nm j q1 y 1 then Bi s 0 for any r g N. r
On the basis of Theorems 3.1, 3.2, and 3.3 we get immediately the following main result of the paper and its corollaries. THEOREM 3.4. Ža.
Let E be a sequence space containing w .
If E has the
SIGNED P
OSCP,
then
E ; F « E ; WF for each Lw-space F. Žb. If E has the
SIGNED P
01-OSCP, then
E ; F « E ; SF for each Lw-space F. In particular, the statements in Ža. and Žb., respecti¨ ely, are fulfilled if F is a separable FK-space, especially, a domain c B of any matrix B. Proof. In light of w1, Theorem 3.6; 4, Theorem 4.4x it is sufficient to prove the theorem in the case of domains c B ; thereby we should note that we may replace in w1, Theorem 3.6x ‘‘separable FK-space’’ by ‘‘Lw-space.’’ Thus, let B be any matrix with E ; c B . Ža. Application of Theorem 3.2 in cases Ži. and Žii. for a proof of E ; LHB for any B with E ; c B . Then, by an inclusion theorem in w3x we get E ; WB for any B. Žb. Application of Ža. and Theorem 3.3. COROLLARY 3.5.
Let E be a sequence space containing w .
Ža. If E has the SIGNED P OSCP, then Ž E b, s Ž E b, E .. is sequentially complete. Žb. If E has the SIGNED P 01-OSCP, then Ž E b, s Ž E b, E .. is sequentially complete and Ž E, t Ž E, E b .. has AK. Proof. See w1; 4, Theorem 3.4x.
533
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
COROLLARY 3.6. Let E be an FK-space containing w and Y be a sequence space ha¨ ing the SIGNED P GHP. Then Y l WE ; F « Y l WE ; WF holds for any Lw-space F. Proof. Theorems 3.1 and 3.4 applied to Remark 2.4Žc.. Deducing consistency theorems is a motivation for summability theorists to examine various gliding hump properties and oscillating properties of sequence spaces. As an immediate corollary of Theorem 3.4 we get a very general consistency theorem containing, with a few exceptions, all known consistency theorems. COROLLARY 3.7 ŽConsistency.. Let E, F be sequence spaces containing w and let A, B be matrices with E q F ; c A l c B . If A and B are consistent on F Ž that is, lim A x s lim B x for each x g F . and E has the SIGNED P OSCP then A and B are consistent on E q F. Proof. Since E has the SIGNED P OSCP we get E ; LHA lLHB Žsee Theorem 3.4 and its proof.. The consistency of A and B on F and w ; F implies lim A e k s a k s bk s lim B e k
Ž k g N. ,
thus lim A x s
Ý ak x k s Ý bk x k s lim B x k
Ž x g E ; LHA lLHB . ;
k
that is, A and B are consistent on E and therefore consistent on E q F. Combining w2, Theorem 1; 4, Theorem 4.4x we get Corollary 3.6 in the case in which Y has the P GHP Žsee also w1, Remark 3.2Ža.x.. To prove that Corollary 3.6 is a strict generalization we show that bs has the SIGNED SP GHP. ŽObviously bs does not have the P GHP.. For that we adjust C. Stuart’s approach to the result that bs has the SIGNED P WGHP Žsee w11, Proposition 3.33x.. THEOREM 3.8. bs has the SIGNED SP GHP and it has neither the Ž for a definition see w1x. nor the ABSOLUTE SP GHP.
P
GHP
534
BOOS, FLEMING, AND LEIGER
Proof. Let x g bs and let Ž y Ž j. . be a block sequence with M [ sup j 5 y Ž j. 5 b ¨ - `. Furthermore, let Žg j . be chosen as in the definition of the block sequence. Since the scalar sequence g jq1 y1
a s Ž aj .
with a j [
Ý
ks g j
y kŽ j. x k
is bounded we may choose a subsequence Ž a jn . g c and we may do it in a way such that < a j y a j < - 2yn n nq r
for all r , n g N.
Ž 11 .
Now we take a subsequence of Ž a jn . and denote it again by Ž a jn .. It fulfills Ž11. too. Choosing the sign sequence Ž hn . s ŽŽy1. nq1 . we get yx g bs, where y s Ýn Žy1. nq1 y Ž jn ., noting the estimations n
Ý
yk x k F
ks1
n
Ý < aj
n
y a j2 i < F 2 iy 1
is1
Ý 2yi is1
if g j 2 nq1 F n - g j 2 nq 1
and Žuse Abel’s partial summation. n
Ý
ks g j2 n y 1 n
Ý
ks g j2 n
yk x k F 3 M 5 x 5 b s
yk x k F 3 M 5 x 5 b s
if g j 2 ny 1 F n - g j 2 n , if g j 2 n F n - g j 2 nq1 y 1.
Thus, bs has the SIGNED SP GHP. The further statements in the theorem are immediate consequences of the fact that x s ŽŽy1. n . g bs. The following result was proved by D. Seydel w9, Satz 3.21x in the case of FK-spaces E containing c 0 and, more generally, by the first and third authors w5x in the case of FK-spaces E containing cs. COROLLARY 3.9.
Let E be an FK-space containing w . Then
Ž X b , s Ž X b , X . . , where X [ bs l WE , is sequentially complete Ž 12. and, equi¨ alently, bs l WE ; F « bs l WE ; WF holds for any Lw-space F.
Ž 13 .
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
535
Proof. From Theorem 3.8, Theorem 3.1, and Remark 2.4Žc., bs l WE has the SIGNED P OSCP. Thus Ž12. follows from Corollary 3.5Ža. and Ž13. from Corollary 3.6. For the equivalence of Ž12. and Ž13. see w4, Theorem 4.4 Ža. m Žd.x. In w11, Proposition 3.30x, C. Stuart gave another proof of the fact that bs has the SIGNED P WGHP. We note that his proof works only if one takes the sign sequence in 1, y14 for real sequences. However, in the real or complex case we can improve the result of C. Stuart}using his method of proof}by the following theorem. THEOREM 3.10. Let Ž m n . be an index sequence with m 0 s 1 and let ny1 Ž y Ž n. . be a block sequence with representation y Ž n. s Ý mjsm y e j such that ny 1 j Ž n. 5 5 sup n y b ¨ \ Q - `. Then for each x g bs there exists a sequence s s Ž s n . in S such that the coordinatewise sum z [ Ý`ns 1 sn xy Ž n. g bs. Proof. Let x g bs and let Ž y Ž n. . be as in the statement of the theorem. Define a signum sequence s by s1 [ sgn² x, y Ž1. :
snq1 [ y
and
sgn² x, y Ž nq1. : sgn Ž Ý nks 1 sk² x, y Ž k . : .
Ž n g N.
with the convention sgnŽ0. s 1 and let z[
`
Ý
sn xy Ž n. .
ns1
Let N g N and choose n g N with m n F N - m nq1; then m ny1
N
Ý
zi F
is1
Ý is1
N
zi q
Ý
zi .
ism n
Now, for all n , m g N, n F m , and all j g N we get by Abel’s partial summation m
Ý
ks n
x k y kŽ j. F 3 5 y Ž j. 5 b ¨ 5 x 5 b s F 3Q 5 x 5 b s \ M,
thus N
Ý ism n
N
zi s
Ý ism n
x i yiŽ nq1. F M.
536
BOOS, FLEMING, AND LEIGER
1y1 < < m ny1 < Clearly <Ý m is1 z i F M and if Ý is1 z i F M then
m nq1 y1
m n y1
zi s
Ý is1
Ý
m nq1 y1
zi q
is1
Ý
zi F M
ism n
since m nq1 y1
Ý ism n
m nq1 y1
zi s
m n y1
x i yiŽ nq1.
Ý
FM
and
ism n
Ý is1
m nq1 y1
zi ,
Ý
zi
ism n
ny1 < have opposite directions. Thus <Ý m is1 z i F M for all n g N and hence N <Ý is1 z i < F 2 M so z g bs. This proves the theorem.
We close the paper with some remarks. Theorem 3.1 tells us that WE has the SIGNED P OSCP whenever E is an FK-space. However, by Theorem 3.4Žb., WE does not have the SIGNED P 01-OSP if E is a separable FK-space fulfilling WE / S E . Following the lines of C. Swartz w12x and C. Stuart w11x it makes sense to generalize the gliding hump and oscillating properties defined in 2.1, 2.2, and 2.3 to spaces of vector valued sequences. In particular, Theorem 3.4Žb. remains true in the case of sequence spaces E over Frechet spaces and ´ domains F of operator valued matrices. By that, we get an essential generalization of the result of C. Stuart w11, Theorem 3.5x. These and other observations dealing with the vector valued case will be presented in another note of the first and third authors.
ACKNOWLEDGMENTS The authors are very thankful to T. Kruth, a student of the first author, and D. Seydel for reading the manuscript very carefully and giving useful hints.
REFERENCES 1. J. Boos and D. J. Fleming, Gliding hump properties and some applications, Internat. J. Math. Sci. 18 Ž1995., 121]132. 2. J. Boos and T. Leiger, General theorems of Mazur]Orlicz type, Studia Math. 92 Ž1989., 1]19. 3. J. Boos and T. Leiger, Consistency theory for operator valued matrices, Analysis 11 Ž1991., 279]292. 4. J. Boos and T. Leiger, Some new classes in topological sequence spaces related to L r-spaces and an inclusion theorem for KŽX.-spaces, Z. Anal. Anwendungen 12 Ž1993., 13]26.
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
537
5. J. Boos and T. Leiger, Weak wedge spaces and theorems of Mazur]Orlicz type, Acta Comm. Uni¨ . Tartuensis 960 Ž1993., 23]28. 6. J. Boos and T. Leiger, The signed weak gliding hump property, Acta Comm. Uni¨ . Tartuensis 970 Ž1994., 13]22. 7. P. K. Kamthan and M. Gupta, ‘‘Sequence Spaces and Series,’’ Dekker, New YorkrBasel, 1981. 8. D. Noll, Sequential completeness and spaces with the gliding humps property, Manuscripta Math. 66 Ž1990., 237]252. 9. D. Seydel, ‘‘Quotienten- und Vertraglichkeitssatze ¨ ¨ fur ¨ Matrizen,’’ Dissertation, Hagen, 1990. 10. A. K. Snyder, Consistency theory in semiconservative spaces, Studia Math. 71 Ž1982., 1]13. 11. C. E. Stuart, ‘‘Weak Sequential Completeness in Sequence Spaces,’’ Thesis, New Mexico State University, Las Cruces, NM, 1993. 12. C. Swartz, The gliding hump property in vector sequence spaces, Mh. Math. 116 Ž1993., 147]158. 13. A. Wilansky, ‘‘Summability through Functional Analysis,’’ Notas de Matematica, Vol. 85, ´ North-Holland, AmsterdamrNew YorkrOxford, 1984.
200, 519]537 Ž1996.
0222
Sequence Spaces with Oscillating Properties Johann BoosU Fachbereich Mathematik, Fernuni¨ ersitat}Gesamthochschule, Postfach 940, D-58084 ¨ Hagen, Germany
Daniel J. Fleming† Department of Mathematics, St. Lawrence Uni¨ ersity, Canton, New York 13617
and Toivo Leiger ‡ ¨ Puhta Matemaatika Instituut, Tartu Ulikool, EE 2400 Tartu, Estonia Submitted by Brian S. Thomson Received February 18, 1994 DEDICATED TO PROFESSOR K. ZELLER ON THE OCCASION OF HIS
70TH BIRTHDAY In this note we consider various types of oscillating properties for a sequence space E being motivated by an oscillating property introduced by Snyder and by recent papers dealing with theorems of Mazur]Orlicz type and gliding hump properties. Our main tools, two summability theorems, allow us to identify two such oscillating properties for a sequence space E one of which provides a sufficient condition for E ; F to imply E ; WF while the other affords a sufficient condition for E ; F to imply E ; SF . Here F is any Lw -space, a class of spaces which includes the class of separable FK-spaces, SF denotes the elements of F having sectional convergence, and WF denotes the elements of F having weak sectional convergence. This, in turn, is applied to yield improvements on some other theorems of Mazur]Orlicz type and to obtain a general consistency theorem. Furthermore, combining the above observations with the work of Bennett and Kalton we obtain the first oscillating property on a sequence space E as a * E-mail: [email protected]. † E-mail: [email protected]. ‡ E-mail: [email protected]. 519 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
520
BOOS, FLEMING, AND LEIGER
sufficient condition for E b, the b-dual of E, to be s Ž E b, E . sequentially complete whereas the second assures both the weak sequential completeness of E b and the AK-property for E with the Mackey topology of the dual pair Ž E, E b .. Q 1996 Academic Press, Inc.
1. INTRODUCTION Numerous authors have used various gliding hump properties imposed on the multiplier space of a sequence space E which are weaker than the assumption that E be solid but are nonetheless sufficient to obtain consistency theorems or, equivalently, to establish the weak sequential completeness of E b, the b-dual of E, and thus getting theorems of Mazur]Orlicz type Žsee w2x for references .. Recent experience Žsee e.g., w8, 12, 11, 1x. has shown that it is sufficient to look for gliding hump properties satisfied for each x in E, that is, ‘‘pointwise’’ gliding hump properties, rather than imposing a uniform hump property on E via the multiplier space M Ž E .. For example, Noll in w8x introduced the so-called weak gliding hump property and showed that it is a sufficient condition for the sequential completeness of the b-dual. Stuart w11x extended Noll’s result for the signed weak gliding hump property. The latter property captures the space bs whereas the former does not. See also w6x for additional results on the signed weak gliding hump property. Furthermore, in w1, 11x Žsee also w6x. it is pointed out that the weak gliding hump property and, more generally, the signed weak gliding hump property of a sequence space E implies not only the weak sequential completeness of E b but also the AK-property of the Mackey topology of the dual pair Ž E, E b .. This proves that there is a big gap between the class of sequence spaces having the signed weak gliding hump property and the class of sequence spaces with weakly sequentially complete b-duals. In earlier papers of the first and third authors it is shown that X [ Y l WE has weakly sequentially complete b-dual when the sequence space Y has a suitable gliding hump property and E is an FK-space containing the finitely non-zero sequences. Here WE denotes the set of all elements of E which are the weak limits of their sections. Although such spaces X have weakly sequentially complete b-duals they will, in general, fail both the signed weak gliding hump property and the AK-property with respect to the Mackey topology of the dual pair Ž X, X b .. Therefore it is of mathematical interest to identify a class of sequence spaces with weakly sequentially complete b-duals which contains both the class of sequence spaces with the signed weak gliding hump property and the class of spaces Y l WE described above. This goal leads us naturally to the class of sequence spaces having so-called oscillating properties.
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
521
Our main tools are Theorems 3.2 and 3.3, which should be of independent interest to summability theorists since they assert that if a matrix B sums a sequence x having suitable properties then it cannot sum a lot of sequences of the type yx, where y is a suitable oscillating sequence. As a consequence we obtain several theorems of Mazur]Orlicz type which improve upon theorems in w2, 6, 1x.
2. NOTATION AND PRELIMINARIES Let v denote the linear space of all scalar Žreal or complex. sequences. By a sequence space E we shall mean any linear subspace of v . A sequence space E with a locally convex topology t is called a K-space if the inclusion map i: Ž E, t . ª v is continuous, where v has the topology of coordinatewise convergence. A K-space with a Frechet topology is ´ called an FK-space. If, in addition, the topology is normable then it is called a BK-space. We will assume throughout familarity with the standard FK-spaces and their natural topologies as well as the properties enjoyed by these spaces Žsee, e.g., w13, 7x.. We refer the reader to w4x for a discussion of Lw-spaces, a class of K-spaces which includes, as a proper subset, the class of separable FK-spaces. For a sequence space E the multiplier space of E and the b-dual of E are given by M Ž E . s x g v < xy g E for each y g E 4 and Eb s x g v
½
Ý x k yk converges for each k
5
ygE ,
where xy denotes the coordinatewise product. Let e denote the sequence of ones and let e k s Ž d jk .`js1 be the kth coordinate vector. For x g v , n g N the nth section of x is n
Pn Ž x . s
Ý
xk ek.
ks1
If Ž E, F . is a dual pair then s Ž E, F ., t Ž E, F . denotes the weak topology and the Mackey topology, respectively. For a sequence space E and a linear subspace F of E b, Ž E, F . is a dual pair under the natural bilinear form ² x, y : s
Ý x k yk . k
522
BOOS, FLEMING, AND LEIGER
If E is a K-space containing w , the space of finitely non-zero sequences, we let WE s x g E < Pn Ž x . ª x Ž s Ž E, EX . . 4 SE s x g E < Pn Ž x . ª x in E 4 , where EX denotes the topological dual of E. A K-space E containing w with E s SE is called an AK-space. If A s Ž a n k . is an infinite matrix with scalar entries the convergence domain
½
c A s x g v < Ax s
`
ž
Ý an k x k k
/
gc ns 1
5
admits a natural FK-topology w13x. For x g c A we write lim A x s lim Ax. If w ; c A let a k s lim n a n k and define
½
IA s x g c A
Ý ak x k exists k
5
,
L A : IA ª K by L AŽ x . s lim A x y Ý k a k x k Žwhere K s C or K s R., and LHA s x g IA < L A Ž x . s 0 4 . Further if w ; c A we write WA , SA instead of Wc A , Sc A . Obviously WA ; LHA . A sequence space E is said to be solid if m ; M Ž E . and monotone if m 0 ; M Ž E ., where m 0 denotes the linear span of all sequences of zeros and ones. In w8x Noll Žsee also w12x. introduced the weak gliding hump property of a sequence space E and showed that if E enjoys this property then E b is s Ž E b, E . sequentially complete. However, the space bs, which is known to have weakly sequentially complete b-dual, fails the weak gliding hump property. Motivated by this observation Stuart w11x extended Noll’s result to spaces which have the signed weak gliding hump property and he showed that bs satisfies the latter property if the scalar field is K s R. Here we introduce several new signed properties with the aim of obtaining improved theorems of Mazur]Orlicz type. A sequence Ž y Ž j. . in v is called a block sequence if there exists an index sequence Žg j . with g 1 s 1 and y kŽ j. s 0 for k - g j and g jq1 F k Ž j g N.. jq 1 y1 y Ž j. e k and the Therefore, each y Ž j. has the representation y Ž j. s Ýgks gj k Ž j. coordinatewise sum y s Ý j y is the limit of this series in Ž v , tv . and is given by y k s y kŽ j. for g j F k - g jq1 and j g N. By definition, a block sequence Ž y Ž j. . is a 1-block sequence if y kŽ j. s 1 Žg j F k - g jq1 and j g N..
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
523
A sequence Ž y Ž j. . in v is called a step 1-block sequence with respect to an index sequence Ž k i . with k 1 s 1 if there exists an increasing sequence Ž i j . in N such that:
Ža.
i 3 jy1 - i 3 j
Ž b.
y kŽ j. s
Žg .
½
Ž j g N. ,
0
if k - k i 3 jy 2 or k G k i 3 jq 1
1
if k i 3 jy 1 F k - k i 3 j
Ž ykŽ j. . is monotonic for k i
3 jy 2
Ž j g N . and y 1Ž1. G 0,
y 1 F k F k i 3 jy 1
and for k i 3 j y 1 F k F k i 3 jq 1
Žd.
ys
Ž j g N. ,
Ý y Ž j. Ž pointwise sum. is constant for k i F k - k iq1 Ž i g N. . j
Furthermore, it is called a step 1-block sequence if there exists an index sequence Ž k i . such that Ž y Ž j. . is a step 1-block sequence with respect to Ž k i .. In particular, each 1-block sequence is a step 1-block sequence and any step 1-block sequence Ž y Ž j. . fulfills 0 F y kŽ j. F 1 Ž j g N. and sup j 5 y Ž j. 5 b ¨ s 2. A 1-block sequence is said to be a 1-block sequence with respect to an index sequence Ž k i . if it is a step 1-block sequence with respect to Ž k i .. The definition of a step 1-block sequence is illustrated by the following picture.
In the following definitions the convergence of the sums is coordinatewise. DEFINITION 2.1. A sequence space E containing w has the signed pointwise gliding hump property ŽSIGNED P GHP. if for each x g E and every block sequence Ž y Ž j. . with sup j 5 y Ž j. 5 b ¨ - ` there exist a subsequence Ž y Ž n k . . of Ž y Ž j. . and a sequence Ž h k . in 1, y14 such that yx g E, where y [ Ý k h k y Ž n k ..
524
BOOS, FLEMING, AND LEIGER
Now other gliding hump properties can be defined depending upon additional properties enjoyed by the subsequence Ž y Ž n k . . in Definition 2.1. Ž2.1.1. If Ž y Ž n k . . can be chosen such that for each subsequence Žy of it there exists a sequence Ž h k . in 1, y14 with yx g E, where y [ Ý k h k y Ž m k . then we say E has the signed strong pointwise gliding hump property ŽSIGNED SP GHP.. Ž2.1.2. If Ž y Ž n k . . can be chosen such that for each subsequence Ž m Ž y k . . of it and each Ž h k . in 1, y14 we get yx g E, where y [ Ý k h k y Ž m k . then we say E has the absolute strong pointwise gliding hump property ŽABSOLUTE SP GHP.. Ž mk..
For convenience we use in the following the notion of a strong subsequence Ž y Ž jk . . of a sequence Ž y Ž j. . in the sense that N R jk < k g N4 is an infinite subset of N. DEFINITION 2.2. Let E be a sequence space containing w . E is defined to have the signed pointwise oscillating property ŽSIGNED P OSCP. if for each x g E and any index sequence Ž k i . with k 1 s 1 there exists a strong subsequence Ž y Ž n j . . of a step 1-block sequence Ž y Ž n . . with respect to Ž k i . such that there exists a sequence Ž h j . in 1, y14 with yx g E, where y [ Ý j h j y Ž n j .. Now similarly to the case of gliding hump properties other oscillating properties can be defined; in the following we need Ž2.2.1. If in Definition 2.2, Ž y Ž n . . can subsequence Ž y Ž n j . . of it and each sequence where y [ Ý j h j y Ž n j ., then we say E has oscillating property ŽABSOLUTE SP OSCP..
be chosen such that for each Ž h j . in y1, 14 we get yx g E, the absolute strong pointwise
We remark that the just defined oscillating properties are motivated by an oscillating property introduced by Snyder w10x and some recent papers connected with theorems of Mazur]Orlicz type and gliding hump properties. DEFINITION 2.3. We say E has the signed pointwise weak gliding hump property ŽSIGNED P WGHP. and the signed pointwise 01-oscillating property ŽSIGNED P 01-OSCP. if the definition of the SIGNED P GHP and the SIGNED Ž j. P OSCP is fulfilled for subsequences of 1-block sequences Ž y ., respectively. Note that the
SIGNED P
WGHP
is due to C. Stuart w11x.
Remark 2.4. Let E and Y be sequence spaces containing w . Ža. Žb. P
ABSOLUTE SP
GHP ; ABSOLUTE SP
ABSOLUTE SP
GHP ; SIGNED SP
WGHP ; SIGNED P
01-OSCP ; SIGNED
OSCP
; SIGNED
GHP ; SIGNED P P
OSCP.
P
OSCP.
GHP ; SIGNED
525
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
Žc. If Y has the SIGNED then E l Y has the SIGNED P
P
GHP
and E has the
ABSOLUTE SP
OSCP
OSCP.
If we replace in the above definitions of the gliding hump and oscillating properties Žsee 2.1, 2.2, and 2.3. the set 1, y14 by S [ z g K < < z < s 14 , where K denotes the scalar field, that is, we take in the case of complex sequences a signum sequence on the unit circle instead of a sign sequence, then we get more general definitions and all results of the paper remain true. 3. MAIN RESULTS First of all we identify large classes of sequence spaces having the GHP and the ABSOLUTE SP OSCP, respectively.
ABSOLUTE SP
THEOREM 3.1. ABSOLUTE SP
GHP
Let E be an FK-space containing w . Then S E has the whereas WE has the ABSOLUTE SP OSCP.
Proof. The first part of the statement can be proved by a refinement of the proof of w1, Theorem 3.3x. For a proof of the second, let x g WE and an index sequence Ž k i . be fixed. Furthermore, let ! ! be a paranorm generating the FK-topology t of E. According to w2, Lemma 1x Žsee also the text following w3, Remark 2.3x. we may choose a sequence Ž x Ž r . . in the convex hull of Pk iy1Ž x . < i g N4 such that in Ž E, t . ,
xŽr. ª x
Ž 1.
tr
xŽr. s
Ý mr i Pk y1Ž x . i
iss r
ž
tr
sr , t r g N with sr F t r - srq1 , 0 F m r i F 1, m r t r / 0,
Ý mr i s 1 iss r
/
. Ž 2.
By virtue of Ž1. we may choose an index sequence Ž r j . such that ! x Ž r . y x Ž rq m . !- 2yjy1
Ž m g N and
r G rj . .
Ž 3.
Then we define z Ž j. [ x Ž r 2 j . y x Ž r 2 jy 1 . Ž j g N .
z[
and
Ý z Ž j. Ž pointwise sum. . j
Because of the representation of x with z Ž j. s y Ž j. x, y kŽ j. s
¡0 ¢1
~
Žr.
in Ž2. we may choose y Ž j. Ž j g N.
if k - k s r if k t r
2 jy 1
2 jy 1
or k G k t r
F k - k sr . 2j
2j
526
BOOS, FLEMING, AND LEIGER
Furthermore, as we may check, Ž y Ž j. . is a step 1-block sequence with respect to Ž k i .. From Ž3. we get that for each subsequence Ž y Ž k j . . of Ž y Ž j. . and each Ž h j . in S the sequence ŽÝ Njs1 h j y Ž k j . x .N is a Cauchy sequence in Ž E, t .. Therefore, because of the completeness of Ž E, t . and the coordinatewise convergence, we have
˜yx g E
with ˜ y[
Ý h j y Ž k . Ž coordinatewise sum. . j
j
Evidently, ˜ y and Ž h j y Ž k j . . satisfy the conditions in Lemma 2 in w2x, thus ˜yx g WE . That completes the proof of the theorem. Note that we have considered in the above proof more general sequences Ž h j . in S . The following ‘‘non-summability theorems’’ form the basis for the main results of this paper and should be of independent mathematical interest. THEOREM 3.2. Let B be a matrix with w ; c B and let x g c B be gi¨ en such that at least one of the following statements Ži., Žii. is fulfilled: ny1 b x / Ži. There exists an index sequence Žhn . such that lim n Ýhks1 k k lim B x. Žii. supn <Ýnks1 bk x k < s `.
Then there exists an index sequence Ž k i . such that for each strong subsequence Ž y Ž m j . . of any step 1-block sequence Ž y Ž m . . with respect to Ž k i . we ha¨ e z [ yx f c B , where y [ Ý j h j y Ž m j . Ž pointwise sum. and Ž h j . is any sequence in S . Proof. First of all we make some considerations in advance. In both case Ži. and case Žii. we will choose an index sequence Ž k i . depending upon x g c B and information resulting from this fact and w ; c B . For all index sequences Ž k i . and Ž n i . and any sequence z g v we use the notations
Ý bn k z k s A i q AUi q Bi q Ci i
Ž i g N. ,
k
where the convergence of Ý k bn i k z k is assumed, k iy1
Ai [
Ý Ž bn k y bk . z k , i
AUi [
ks1
Ý ksk i
Ý
bk z k ,
ks1
k iq1 y1
Bi [
k iy1
bn i k z k
and
Ci [
`
Ý ksk iq1
bn i k z k .
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
527
In all cases we will construct the index sequences Ž k i . and Ž n i . such that
Ž A i . g c0
and
Ž Ci . g c0 ,
where z is chosen as is described in the theorem. It is then immediate that each of the following conditions implies z f c B : Ž a . Ž AUi . g c and Ž Bi . f c. Ž b . Ž AUi . f c and Ž Bi . g c 0 , where Ž i j . is a suitable index sequence. j j Let x g c B and let Žhn . with h1 s 1 be any given index sequence. ŽLater on we will fix Žhn . on the base of the properties Ži. and Žii... Since w ; c B and x g c B we may choose index sequences Ž n i ., Ž k i ., and Ž n i . having certain properties: For n 1 [ 1 and k 1 [ hn 1 we may choose an n1 g N such that k 1y1
Ý
< bn k y bk < < x k < - 2y1
Ž n G n1 . .
ks1
Then we may choose n 2 ) n 1 such that for k 2 [ hn 2 L
Ý bn k x k
- 2y2
Ž n F n1 , k 2 F l - L . .
ksl
If we have chosen n iy1 and n i then for k i [ hn i we determine n i ) n iy1 with k iy1
Ý
< bn k y bk < < x k < F 2yi
Ž n G ni . ;
Ž 4.
ks1
furthermore, we choose n iq1 ) n i such that for k iq1 [ hn iq 1 we have L
Ý bn k x k
- 2yŽ iq1.
Ž n F n i , k iq1 F l - L . .
Ž 5.
ksl
Now, let Ž y Ž m . . be any step 1-block sequence with respect to Ž k i .. Then we consider in both cases strong subsequences Ž y Ž m j . . of Ž y Ž m . .. Using 5 y 5 ` F 1 for y [ Ý j h j y Ž m j ., where h j g S Ž j g N., and noting Ž d . in the definition of a step 1-block sequence we get for z [ yx the estimations k iy1
Ý ks1
< bn k y bk < < x k < i
iª`
0
6
< Ai < F
528
BOOS, FLEMING, AND LEIGER
because of Ž4. and `
`
k rq1 y1
Ý
Ý
rsiq1
ksk r
bn i k z k F
Ý ksk iq1
`
bn i k x k F
2yr
Ý
iª`
0
6
< Ci < F
rsiq1
because of Ž5.. Now, we shall fix Žhn . depending upon Ži. and Žii.. In case Ži. we may choose Žhn . such that h1 s 1 and
a [ lim n
hny1
Ý
bk x k / lim B x \ d;
ks1
furthermore we may assume hnq m y1
bk x k - 2yn
Ý
ks hn
Ž n , m g N. .
Ž 6.
For this Žhn . let Ž k i ., Ž n i ., and Ž n i . be chosen as is described above. Furthermore, let Ž y Ž m . . be any step 1-block sequence with respect to Ž k i . and let Ž y Ž m j . . be any strong subsequence of Ž y Ž n . .. Since it is strong there exists an index sequence Ž jr . with m j r q 1 / m j rq1 for each r g N. Roughly speaking, if we consider y [ Ý j y Ž m j . Žcoordinatewise sum. then there exists a 0-block in between y Ž m jr . and y Ž m jrq 1 . ; in particular, if Ž i j . is chosen as in the definition of a step 1-block sequence, then we have yk s
¡0 ¢1
~
if k - k i 3 m
or k i 3 m
1y 2
if k i 3 m y 1 F k - k i 3 m j
j rq 1
F k - k i3m
2 j rq 1 y
Ž r g N. Ž 7.
Ž j g N.
j
as we may verify with Ž b . in the definition of a 1-block sequence by the aid of the picture following that definition. We will prove z [ yx f c B , where y [ Ý j h j y Ž m j . and h j g S Ž j g N.. Let r , l g N with 1 F l - r be given. By that and Ž6. we get kry1
Ý ks1
bk z k y
kl y1
Ý
k r y1
bk z k s
ks1
bk z k
Ý kskl
nry1 hm q1 y1
Ý
ms nl
Ý
ks hm
bk x k F
implying the existence of k iy1
lim i
Ý ks1
bk z k
Ý
m s nl
2ym
lª`
0,
6
F
nr y1
529
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
and thus Ž AUi . g c. Now, we prove Ž Bi . f c. By construction we know that zk s
¡h x ¢0
~
j
if k i 3 m y 1 F k - k i 3 m
k
j
if k i 3 m
F k - k i3m
j rq 1
Ž j g N.
j
Ž r g N. .
j rq 2
Thus Bi 3 m s 0 by Ž7.; therefore Ž Bi 3 m . g c 0 . Then Ž Bi . f c is proved j rq 1 j rq 1 if Ž Bi 3 m . f c 0 . However, if one puts i [ i 3 m j y1 , this follows from the j ry 1
r
identities k iq1 y1
Bi s
k iq1 y1
bn i k z k s h j r
Ý ksk i
s h jr
ž
bn i k x k
Ý ksk i
k iy1
Ý bn k x k y Ý i
k
k iy1
/
bk x k y h jr
ks1
`
Ý Ž bn k y bk . x k y h j Ý i
r
ks1
bn i k x k
ksk iq1
iy1 b x . because of Ž4. and Ž5. and the fact that ŽÝ`ks 1 bn i k x k y Ý kks1 k k igN U converges to d y a / 0. Altogether, we proved Ž A i . g c and Ž Bi . f c, thus z f c B by Ž a .. In case Žii., without loss of generality, we may assume
sup R n
ž
n
Ý bk x k ks1
/
s `.
Therefore, we may choose an index sequence Žhn . such that h1 s 1 and R
ž
hnq1 y1
Ý
ks hn
hn y1
/
bk x k G n q
Ý
< bk x k <
Ž n g N. .
Ž 8.
ks1
Again, for this Žhn . let Ž k i ., Ž n i ., and Ž n i . be chosen as is described above. Furthermore, let Ž y Ž m . . be any step 1-block sequence with respect to Ž k i ., let Ž y Ž m j . . be any strong subsequence of Ž y Ž m . ., and let Ž y Ž m jr . . be chosen as in case Ži.. We will prove z [ yx f c B , where y [ Ý j h j y Ž m j . and h j g S Ž j g N.. Considering i [ i 3 m j q1 we get by Ž7. and the definition of z the r statement Bi s 0, thus Ž Bi . g c 0 , and also Ž AUi . f c since <
AUi <
k iy1
s
Ý
k i 3m
bk z k G < h j r <
ks1
y1
k i 3m y1 jr
ksk i 3m
0
bk x k y
Ý j ry 1
bk x k y kŽ m j r . y
Ý ksk i 3m
k i3 m
j ry 1
y1
bk z k
Ý ks1
j ry 1
k i3 m
j ry 1
y1
Ý ks1
< bk z k < G 3 m j
rª` r
`,
6
GR
j rq 1
530
BOOS, FLEMING, AND LEIGER
where we made use of R
ž
hnq1 y1
Ý
ks hn
/
Ž n g N. ,
bk x k G 0
which follows from Ž8.. Altogether we have shown z f c B by Ž b . and statement Ža. is proved. THEOREM 3.3. Let B be a matrix with w ; c B and let x g c B R SB be gi¨ en. Then there exists an index sequence Ž ln . such that for each strong subsequence Ž y Ž m j . . of any 1-block sequence Ž y Ž m . . with respect to Ž ln . we ha¨ e z [ yx f c B , where y [ Ý j h j y Ž m j . Ž pointwise sum. and Ž h j . is any sequence in S . Proof. We assume x g LHB RSB Žotherwise we may apply Theorem 3.2.. First of all we remark that x f SB if and only if there exists an « ) 0 such that for each n g N there exist index sequences Ž nn j . j and Ž bn j . j with bn jy1 < bn 1 ) n such that <Ý ks n bnn j k x k G « . Let « ) 0 and let the index sequences be chosen in that way. We put k 1 [ 1 and choose n1 , k 2 g N with k 2 ) k 1 such that 1
Ý < bn k y bk < < x k < - 2y1
Ž n G n1 .
ks1
and k 2y1
bn1 k x k G « .
Ý ksk 1
After that we may choose n 2 , k 3 g N with k 3 ) k 2 and n 2 ) n1 such that k 2y1
Ý
< bn k y bk < < x k < - 2y2
Ž n G n2 . ,
ks1 k 3y1
Ý
bn 2 k x k G «
ksk 2
and L
Ý bn k x k ksl
- 2y2
L
and
Ý bk x k ksl
- 2y2
Ž n F n1 , k 3 F l F L . .
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
531
Having fixed n1 , . . . , n i and k 2 , . . . , k iq1 we choose n iq1 , k iq2 g N with k iq2 ) k iq1 and n iq1 ) n i such that k iq1 y1
Ý
< bn k y bk < < x k < - 2yŽ iq1.
Ž n G n iq1 . ,
ks1 k iq2 y1
bn iq 1 k x k G «
Ý
Ž 9.
ksk iq1
and L
L
- 2yŽ iq1.
Ý bn k x k
- 2yŽ iq1.
Ý bk x k
and
ksl
ksl
Ž n F n i , k iq2 F l F L . . Now, we put ln [ k 2 ny1 Ž n g N. and let Ž y Ž m . . be a 1-block sequence with respect to Ž ln . and Ž y Ž m j . . be a strong subsequence of Ž y Ž m . .. Since it is a strong subsequence there exists an index sequence Ž jr . with m j r q 1 / m j rq1 Ž r g N.. For a representation of y Ž m . Ž m g N. we may choose an index sequence Ž nm . with y kŽ m . s
½
1
if lnm F k - lnm q 1
0
otherwise
Ž m , k g N. .
Thus, for y [ Ý j h j y Ž m j . Žpointwise sum., where Ž h j . is any sequence in S , we have yk s
¡h ¢0
~
if lnm s k 2 nm y1 F k - lnm q 1 s k 2 nm q 1y1
j
j
j
if k - lnm or lnm 1
j
j rq 1
j
F k - lnm
j rq 1
.
Ž 10 .
Using the notations for A i and AUi as in the proof of 3.2 and k iq2 y1
Bi [
bn i k z k
Ý
and
Ci [
ksk i
`
bn i k z k
Ý ksk iq2
we get 0,
Ž AUi . g c,
Ci
iª`
0
6
iª`
6
Ai
for z [ yx quite similarly to the corresponding cases in the proof of 3.2.
532
BOOS, FLEMING, AND LEIGER
Thus, z f c B is proved if Ž Bi . f c. But this follows by the following considerations: If r g N and i [ 2 nm j q1 y 2 then by Ž9. and Ž10. r
< Bi < s
k iq1 y1
Ý
bn i k z k s < h j <
ksk i
k iq1 y1
Ý
bn i k x k G « ) 0
ksk i
and if i [ 2 nm j q1 y 1 then Bi s 0 for any r g N. r
On the basis of Theorems 3.1, 3.2, and 3.3 we get immediately the following main result of the paper and its corollaries. THEOREM 3.4. Ža.
Let E be a sequence space containing w .
If E has the
SIGNED P
OSCP,
then
E ; F « E ; WF for each Lw-space F. Žb. If E has the
SIGNED P
01-OSCP, then
E ; F « E ; SF for each Lw-space F. In particular, the statements in Ža. and Žb., respecti¨ ely, are fulfilled if F is a separable FK-space, especially, a domain c B of any matrix B. Proof. In light of w1, Theorem 3.6; 4, Theorem 4.4x it is sufficient to prove the theorem in the case of domains c B ; thereby we should note that we may replace in w1, Theorem 3.6x ‘‘separable FK-space’’ by ‘‘Lw-space.’’ Thus, let B be any matrix with E ; c B . Ža. Application of Theorem 3.2 in cases Ži. and Žii. for a proof of E ; LHB for any B with E ; c B . Then, by an inclusion theorem in w3x we get E ; WB for any B. Žb. Application of Ža. and Theorem 3.3. COROLLARY 3.5.
Let E be a sequence space containing w .
Ža. If E has the SIGNED P OSCP, then Ž E b, s Ž E b, E .. is sequentially complete. Žb. If E has the SIGNED P 01-OSCP, then Ž E b, s Ž E b, E .. is sequentially complete and Ž E, t Ž E, E b .. has AK. Proof. See w1; 4, Theorem 3.4x.
533
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
COROLLARY 3.6. Let E be an FK-space containing w and Y be a sequence space ha¨ ing the SIGNED P GHP. Then Y l WE ; F « Y l WE ; WF holds for any Lw-space F. Proof. Theorems 3.1 and 3.4 applied to Remark 2.4Žc.. Deducing consistency theorems is a motivation for summability theorists to examine various gliding hump properties and oscillating properties of sequence spaces. As an immediate corollary of Theorem 3.4 we get a very general consistency theorem containing, with a few exceptions, all known consistency theorems. COROLLARY 3.7 ŽConsistency.. Let E, F be sequence spaces containing w and let A, B be matrices with E q F ; c A l c B . If A and B are consistent on F Ž that is, lim A x s lim B x for each x g F . and E has the SIGNED P OSCP then A and B are consistent on E q F. Proof. Since E has the SIGNED P OSCP we get E ; LHA lLHB Žsee Theorem 3.4 and its proof.. The consistency of A and B on F and w ; F implies lim A e k s a k s bk s lim B e k
Ž k g N. ,
thus lim A x s
Ý ak x k s Ý bk x k s lim B x k
Ž x g E ; LHA lLHB . ;
k
that is, A and B are consistent on E and therefore consistent on E q F. Combining w2, Theorem 1; 4, Theorem 4.4x we get Corollary 3.6 in the case in which Y has the P GHP Žsee also w1, Remark 3.2Ža.x.. To prove that Corollary 3.6 is a strict generalization we show that bs has the SIGNED SP GHP. ŽObviously bs does not have the P GHP.. For that we adjust C. Stuart’s approach to the result that bs has the SIGNED P WGHP Žsee w11, Proposition 3.33x.. THEOREM 3.8. bs has the SIGNED SP GHP and it has neither the Ž for a definition see w1x. nor the ABSOLUTE SP GHP.
P
GHP
534
BOOS, FLEMING, AND LEIGER
Proof. Let x g bs and let Ž y Ž j. . be a block sequence with M [ sup j 5 y Ž j. 5 b ¨ - `. Furthermore, let Žg j . be chosen as in the definition of the block sequence. Since the scalar sequence g jq1 y1
a s Ž aj .
with a j [
Ý
ks g j
y kŽ j. x k
is bounded we may choose a subsequence Ž a jn . g c and we may do it in a way such that < a j y a j < - 2yn n nq r
for all r , n g N.
Ž 11 .
Now we take a subsequence of Ž a jn . and denote it again by Ž a jn .. It fulfills Ž11. too. Choosing the sign sequence Ž hn . s ŽŽy1. nq1 . we get yx g bs, where y s Ýn Žy1. nq1 y Ž jn ., noting the estimations n
Ý
yk x k F
ks1
n
Ý < aj
n
y a j2 i < F 2 iy 1
is1
Ý 2yi is1
if g j 2 nq1 F n - g j 2 nq 1
and Žuse Abel’s partial summation. n
Ý
ks g j2 n y 1 n
Ý
ks g j2 n
yk x k F 3 M 5 x 5 b s
yk x k F 3 M 5 x 5 b s
if g j 2 ny 1 F n - g j 2 n , if g j 2 n F n - g j 2 nq1 y 1.
Thus, bs has the SIGNED SP GHP. The further statements in the theorem are immediate consequences of the fact that x s ŽŽy1. n . g bs. The following result was proved by D. Seydel w9, Satz 3.21x in the case of FK-spaces E containing c 0 and, more generally, by the first and third authors w5x in the case of FK-spaces E containing cs. COROLLARY 3.9.
Let E be an FK-space containing w . Then
Ž X b , s Ž X b , X . . , where X [ bs l WE , is sequentially complete Ž 12. and, equi¨ alently, bs l WE ; F « bs l WE ; WF holds for any Lw-space F.
Ž 13 .
SEQUENCE SPACES WITH OSCILLATING PROPERTIES
535
Proof. From Theorem 3.8, Theorem 3.1, and Remark 2.4Žc., bs l WE has the SIGNED P OSCP. Thus Ž12. follows from Corollary 3.5Ža. and Ž13. from Corollary 3.6. For the equivalence of Ž12. and Ž13. see w4, Theorem 4.4 Ža. m Žd.x. In w11, Proposition 3.30x, C. Stuart gave another proof of the fact that bs has the SIGNED P WGHP. We note that his proof works only if one takes the sign sequence in 1, y14 for real sequences. However, in the real or complex case we can improve the result of C. Stuart}using his method of proof}by the following theorem. THEOREM 3.10. Let Ž m n . be an index sequence with m 0 s 1 and let ny1 Ž y Ž n. . be a block sequence with representation y Ž n. s Ý mjsm y e j such that ny 1 j Ž n. 5 5 sup n y b ¨ \ Q - `. Then for each x g bs there exists a sequence s s Ž s n . in S such that the coordinatewise sum z [ Ý`ns 1 sn xy Ž n. g bs. Proof. Let x g bs and let Ž y Ž n. . be as in the statement of the theorem. Define a signum sequence s by s1 [ sgn² x, y Ž1. :
snq1 [ y
and
sgn² x, y Ž nq1. : sgn Ž Ý nks 1 sk² x, y Ž k . : .
Ž n g N.
with the convention sgnŽ0. s 1 and let z[
`
Ý
sn xy Ž n. .
ns1
Let N g N and choose n g N with m n F N - m nq1; then m ny1
N
Ý
zi F
is1
Ý is1
N
zi q
Ý
zi .
ism n
Now, for all n , m g N, n F m , and all j g N we get by Abel’s partial summation m
Ý
ks n
x k y kŽ j. F 3 5 y Ž j. 5 b ¨ 5 x 5 b s F 3Q 5 x 5 b s \ M,
thus N
Ý ism n
N
zi s
Ý ism n
x i yiŽ nq1. F M.
536
BOOS, FLEMING, AND LEIGER
1y1 < < m ny1 < Clearly <Ý m is1 z i F M and if Ý is1 z i F M then
m nq1 y1
m n y1
zi s
Ý is1
Ý
m nq1 y1
zi q
is1
Ý
zi F M
ism n
since m nq1 y1
Ý ism n
m nq1 y1
zi s
m n y1
x i yiŽ nq1.
Ý
FM
and
ism n
Ý is1
m nq1 y1
zi ,
Ý
zi
ism n
ny1 < have opposite directions. Thus <Ý m is1 z i F M for all n g N and hence N <Ý is1 z i < F 2 M so z g bs. This proves the theorem.
We close the paper with some remarks. Theorem 3.1 tells us that WE has the SIGNED P OSCP whenever E is an FK-space. However, by Theorem 3.4Žb., WE does not have the SIGNED P 01-OSP if E is a separable FK-space fulfilling WE / S E . Following the lines of C. Swartz w12x and C. Stuart w11x it makes sense to generalize the gliding hump and oscillating properties defined in 2.1, 2.2, and 2.3 to spaces of vector valued sequences. In particular, Theorem 3.4Žb. remains true in the case of sequence spaces E over Frechet spaces and ´ domains F of operator valued matrices. By that, we get an essential generalization of the result of C. Stuart w11, Theorem 3.5x. These and other observations dealing with the vector valued case will be presented in another note of the first and third authors.
ACKNOWLEDGMENTS The authors are very thankful to T. Kruth, a student of the first author, and D. Seydel for reading the manuscript very carefully and giving useful hints.
REFERENCES 1. J. Boos and D. J. Fleming, Gliding hump properties and some applications, Internat. J. Math. Sci. 18 Ž1995., 121]132. 2. J. Boos and T. Leiger, General theorems of Mazur]Orlicz type, Studia Math. 92 Ž1989., 1]19. 3. J. Boos and T. Leiger, Consistency theory for operator valued matrices, Analysis 11 Ž1991., 279]292. 4. J. Boos and T. Leiger, Some new classes in topological sequence spaces related to L r-spaces and an inclusion theorem for KŽX.-spaces, Z. Anal. Anwendungen 12 Ž1993., 13]26.
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5. J. Boos and T. Leiger, Weak wedge spaces and theorems of Mazur]Orlicz type, Acta Comm. Uni¨ . Tartuensis 960 Ž1993., 23]28. 6. J. Boos and T. Leiger, The signed weak gliding hump property, Acta Comm. Uni¨ . Tartuensis 970 Ž1994., 13]22. 7. P. K. Kamthan and M. Gupta, ‘‘Sequence Spaces and Series,’’ Dekker, New YorkrBasel, 1981. 8. D. Noll, Sequential completeness and spaces with the gliding humps property, Manuscripta Math. 66 Ž1990., 237]252. 9. D. Seydel, ‘‘Quotienten- und Vertraglichkeitssatze ¨ ¨ fur ¨ Matrizen,’’ Dissertation, Hagen, 1990. 10. A. K. Snyder, Consistency theory in semiconservative spaces, Studia Math. 71 Ž1982., 1]13. 11. C. E. Stuart, ‘‘Weak Sequential Completeness in Sequence Spaces,’’ Thesis, New Mexico State University, Las Cruces, NM, 1993. 12. C. Swartz, The gliding hump property in vector sequence spaces, Mh. Math. 116 Ž1993., 147]158. 13. A. Wilansky, ‘‘Summability through Functional Analysis,’’ Notas de Matematica, Vol. 85, ´ North-Holland, AmsterdamrNew YorkrOxford, 1984.