Series I : COMMENT ATI ONES MATHEMATICAE X V III (1974) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I : PRACE MATEM AT Y CZNE X V III (1974)
H. S amplawski (Gdansk)
On bilinear series in Banach spaces II
We recall the notation given in the first part of paper [2]. The letters X , Y , Z stand for Banach spaces and U: 1 x Г a Z is a (bounded) bilinear operator. 9Л- , SCR0, 9J1+ denote the sets of all monotone sequences у — {^n}w=0)1>... of positive numbers such that \\myn = 0 , /лп = const, lim//n = oo, respectively; we also denote 50to+ =9Л°и9Л+ and 9Л
и m 0+.
Furthermore, if , u 2, ... are elements of a Banach space, let us denote by u. the sequence {un}n=lt2,... ? by An(u.)(ov Anu.) thew-th difference un —un+\i by sn{u.) the n-th partial sum щ + ... -\-un and by r (и .) the sum of remaining terms un+i + un+2 + ... (provided that the series £ u n is convergent). Moreover, let s0(u.) = 0 and r0(u.) = и1 + и2+ ...
We begin by generalizing some concepts investigated in [2] to the case of vector valued sequences.
1. The space A ^ X ). If [леШ ~, then A ^ X ) will denote the set of all sequences x. = of elements of X such that
oo
p( x -) = y n\\An%A\< °°-
n = 1
If /4 6 Ш10+, then Ap(X) will denote the set of all x. for which p {x.) < oo
and such that xn -> 0. If X is the field В of reals, we shall write A^ instead of Afi(B). It is clear that A ^ X ) is a linear space. The norm in А^(Х) is defined as [|ж.|| = / t40||æ1||+p((r.) if /иеАЯ~ and ||ж.|| = р(ж.) if /4еЗЛ0+. I t can be easily verified that A ^ X ) is a Banach space.
If /иеШ, let Г^( Y) denote the set of all sequences y. = {yn} of elements of Y for which ||«n(y.)|| = 0(jun) if /4€TR0+ and \\rn(y.)\\ = 0(,un) if уеШ Г.
Taking for y. in r /i(Y)
Ily.ll = sup И*(зМН if fln
lly.ll = sup ■ \K(y-)\\ if и e 9Л~
Уп
we get also Banach spaces.
7 8 H . S a m p l a w s k i
L e m m a 1 . Let р ч Ш and let the operator A n : X - > A ^ X ) be defined by the equality A n{x) = (0, . . . , a?, 0, ...) fo r n = 1, 2, ... Then fo r each
oo тГ
x. in Л^(Х) the series £ A n(xn) is A^(X)-convergent to x ..
71 — 1
P ro o f. Let u. = {un}, where un = A n(xn), n = 1, 2, ... Then 11ж* sn{u.)\\ = 11(0, . . . , жи+1? xn+2, ...)||
OO
Уп ll^n+lil “l- >
k = n + 1
hence (cf. [2], Lemmas 2 and 4) sn(u.) x ..
T h e o r e m 1. Every continuous linear functional f on Л1г(Х) is o f the form
OO
(1) /0»-) = £<Pn(Xn),
n =
1
where cp. = {рп}е Г /г(Х*) and | J / | | == Ц 9 9 . Ц .
P ro o f. If /e A *(X ) and cpn = A * { f) {A*n denotes the operator adjoint to A n), then by Lemma 1 we get (1).
Let /г€ ЭД0+. Then ||«n(ÿ>.)|| < \\f\\pn and therefore \\<p. ||< ||/||, i. e.
<p.€ Г ^ Х * ). Conversely, if ф.е Г ^ Х * ), then in virtue of Theorem 1 of [ 2 ] , ( 1 ) is a linear functional опА ^ (Х ) and ||/|| < Ц 99 .Ц. If ^е9Л~, according to the inequality
II 9 W 1 + ••• + = SUP |/M»+1+ ••• + А н-лИ |< \\f\\(Pn + f*n+k)
IWK1
we get Ц 99 .И < ||/!|. The inverse inequality follows by Theorem 2 'o f [ 2 ].
L e m m a 2. Let <pn€ X , n — 1 ,2 , ... and let /иеШ. Then the condition (2) fo r every x. in Л^(Х) the series E cpn{xn) is convergent,
is equivalent to the relation:
9 9. e Г /Л(Х*).
P ro o f. If (p.€ Г Ц(Х*), (2) follows by Theorem 1 or 2 of [2] according as fj, e 9Л0+ or Conversely, if (2) holds, we denote
/m (® 0 = <Pl(Xl) + • • • +<Pm(®m) if ^ € 9Л° +
and
f m . y l À ' V ’ ) =
^ f t + l (* ^ * +1 ) "b • * • “b ^ / / e 9 Л ,
where m = 1, 2, ... ; к = 0, 1, ... and к < m. Each of f m, f nhk is a contin
uous linear functional on Л ^ Х ). In virtue of Theorem 1,
\\fmW = max HMffOll Bn
ur 11 _ I Il99/ c + l + - - - + ^ 1 1 ll95A + 2 + - - - + Ç ’jrall l l ^ m l l \
ll/m, к I1 max < , j ... у > .
I Bk Bk+1 В т — 1 J
If //с9Л0+, assumption (2) and the Banach-Steinhaus theorem give SUP ll/да II < °°, i- e- Гр (X*). Similarly, if /леЩ~, then sup||/m fc|| = К
д а к, д а
< оо, hence Il 99 *+1+ ... + 9 ?m|| < K y k for each к and m ( k < m). The series 2<pn is norm convergent in X * and 9 ?.e Гр(Х *).
T h e o r e m 2 . Let у е Ш , let yne Y fo r n = 1 , 2, ... and let the continuous bilinear operator Ü: X x T -> Z satisfy the following condition :
(3) there is an x0 in X (||a?0|| = 1), such that the set {y: \\U(oc0, y)\\ ^ 1} is bounded.
Moreover, i f
(4) fo r every x. in AM(X) the series U(xn, yn) is convergent in Z, then у . = {yn}€ Гр{Х ).
P ro o f. Given a fixed integer n, we denote Vn{x) — U(x, yn). Then Vn : X -> Z is bounded and linear. Let f e Z * and let q>n = V *(f). Since Pn(xn) = f(U (x n, yn)), according to (4) the series £<pn(ocn) is convergent for x. in Ap{X). By'L em m a 2, <p. = {<pn} e i. e. \\8n{<p.)\\ = 0 { y n) if ^€9Л0+ and \\rn{(p.)\\ = 0 ( y n) if уеШ ~ . To consider the case / ae30Io+, we denote Wn{x) = U(x, y~ l sn{y,)). Then
sup П
I M f f Q I B n
— sup sup n ||x||=l
$n(<P-)X Bn
= sup sup \f(wn(x))
n 11*11=1
= sup sup |В7*(/)ж| = sup \\W* (/)||.
n 11*11 = 1 n
for any / in Z*, hence by equality |jltT*|| = ||1YJ| and Banach-Steinhaus theorem, sup||JYJ| = K < 00. Since || TJ[x, Bnlsn(y>))\\ ^ К I N I f ° r x i n X ,
П
condition (3) implies y .e F ^ Y ). If ^еЭЯ- , one has to take the sequence in place of { Wn), where
W tn, к У к+ 1 + • • • + Ут Bk
m = 1 , 2, . . . , к — 0, 1, . . . , к < т .
80 H . S a m p l a w s k i
K e m a rk . If U: X x Y Z is a, continuous bilinear operator, then
OO
0-. Л„(Х) x F J Y) -> Z, where TJ(x.,y.) = £ U(xn, y n) is bilinear and 71= 1
$ \\m if ^ a w °+
( WUWa^ if у еШ ~ 1 where
f*n-1 I
The easy proof is based on Theorems 1 and 2 of [2]. If we apply the preceding considerations not to the pair Л^(Х), Г ц( Y) bnt to the pair of spaces Л^Д, (X)), Y)), y, ve 9Л, we obtain results similar to those of Theorem 1 and 2 of [2] as well as to Theorems 1 and 2 given above,
OO
bnt for the interated bilinear series Ü(xk., yk.), where xk. — {xkn}eA v{X),
k=l
yk. = {ykn}e r v(Y). We omit the details.
2. Some elementary relations for the space A ^ X ) . Given two Banach spaces X and Y, X ~ Y (X ~ Y) means that X and Y are isomorphic (isometricaly isomorphic, i. e. equivalent). Let lp (X), where oo, denote the space of all sequences u. = in X for which
( oo \ i
Ip^ \KWP < 00
n = l /
and
ll^-ll = S U p IK I I < OO
П
(i) I f y e ff î , then Лц(Х) ~ 1 х(Х).'
P ro o f. Let /a«rim0+, let x. = {xn}e Л^(Х) and let
if 1 < p < oo,
i f p = oo.
(5) un = /ик (я> - x n+1), n — 1, 2, ...
Then u .e lx(X) and \\u.\\ — ||ж.||. For a given u. in lx{X), the infinite system of equations (5) has a unique solution x. in Л^(Х). Indeed, by (5), we see that x. = {a?n} is determined with an error not exceeding the first term x x\
xn-\-\ — x\ I Y • • • Y I ? ^ — 2, 3, ...
\ Уп !
Since ^еЗЛ04" and u . e l 1(X), we have ^\\un\\[yn < oo. Taking xx —
00
= 2 u j y n we get p(a?.)< oo, xn -> 0, i. e. х .е Л ^ Х ) and ||ж.|| = Цад.Ц.
n=1
If the correspondence
(xx, x 2 ) .. •) (yoxi ) A*i (x i x f) ? • • •) is a linear isometry from A ^ X ) onto lx(X).
(ii) I f у еШ , then Г^{Х) m l 00(X).
P ro o f. Let x .e Г (X). The correspondences
( rp rr*
^
V*"! ? *^2 ? • • •;
Х л
xx + хг and
{•fin fi%i • • •)
Bi
%1 “Ь #2 ~b • • • Bo
B2 x 2 + x3
Bi
if /иеШ0+
if ^еЭДГ
are the desired isometries from Г р(Х) onto lœ{X).
(iii) l*(X )
Indeed, Theorem 1 and (ii) yield Л *(Х ) ~ Г^(Х*) ~ Z00(X*). Accord
ing to (i), we have Л *(Х ) ~ l * ( X ) .
Let <X x Y ; || • Ц^) denote the Cartesian product of Banach spaces X and T normed as follows:
Ц^У)\\ = (iw r + l& n 1'1’
max {||*||, Hyll}
if 1 < p < oo, if p = oo.
All these spaces are isomorphic.
(iv) I f X ( Y), then Лц{Х) ~ Лр{ Y) but not conversely. Sim ilar statement holds fo r the space Г ^ Х ).
P ro o f. Let Ü be an isomorphism from X onto Y, let х .е Л и(Х) and let y. = {U (xnj}. The correspondence x .- + y . is the desired isomorphism.
Indeed, if for instance /иеШ0+, then \\y.\\ < ||?7||||ж|| and yn -> 0 as n -> o o ,
hence y .t Л^( Y). I t is obvious, that for any y. in Ap(Y ) there is only one x. in Лц(Х) such that V (xn) = уn, n = 1, 2, ... Now we apply the Banach’s inversion theorem. To prove the second part of our statement, we notice that for any two Banach spaces X and Y,
^ « I x Y', II-Hi» ^ < Л „ (Х )х Л „ (У ); ||■ 11 j>•
Taking X = Y = В we get
Л , < - В х й ; I l - I k ) — (At* X Л д ; H - l l j ) c ^ l . x l , ~ ^ ~ A M, but the spaces В x В and В are non-isomorphic.
(v) Let цп = n llP, vn = n m , n = 1, 2, . . . , 1 < p < i/e P = 1.
Then Л„(Х) c lp {X) c r v(X).
This is an easy consequence of Holder’s inequality and the inequality
71 =
1
1( Up
'\p I <
^ n llp\\Anx.\\,
which can be proved similar as in example 3 of [2].
6
— R oczniki PTM — P r a c e M atem atyczn e X V III.
82 H . S a m p l a w s k i
3. The discusion of (B )-condition. Assumption (3) given in Theorem 2 requires some explanations. We shall say that a bilinear operator U: X x
x Y - > Z satisfying condition (3) fulfils the ( В )-condition with respect to y ; the map U need not be symmetric, consequently we define also the (B)-condition with respect to x :
there is an y 0 in Y (||y0|| = 1) such that {xe X : \\U(x, y0)|| < 1} is bounded.
The s i m p l e s t example of a bilinear operator satisfying (B)-condition with r e s p e c t to both variables is obtained if one of the spaces X or Y is
t h e f i e l d of scalars and if U (x, y) — xy. We also observe that if dim, A > 1, then the operator U: X x X * -> B , where U (x,x*) = x * ( x ) , satisfies the (B)-condition neither with respect to x nor to x*, but in spite of this, the statement of Theorem 2 remains true — this follows by Lemma 2.
L
e m m a3. Let U : X x Y -> Z be a bounded bilinear operator and let x 0e X (||гс0|| == 1). The following conditions are equvalent:
(i) the set {ye Y : ||?7(ж0, y)\\ < 1} is bounded,
( n ) 1 Ы 1 о = \\U(x0, y)\\ is a norm in Y and ||-|| ~ II J o -
P ro o f, (ii) => (i) is obvious. If (i), then the functional ||y||0 = \\U(x0,y)\\
is a norm in Y and ||y||0 < ||?7|| ||y||. Moreover, the unit ball in < Y ; ||*||0>
is a bounded set in < Y ; ||*||>, hence there is a positive constant К such that if ||y||0 < 1, then ||y|| < K . Thus, it у ^ 0 , then ||ll2/IIA12/||o = 1* Hence
||ll2/llo- 1 2/|| < ^ i-e. 112/11 < X ||y||0.
By Lemma 3 we observe that if the set { y * Y : \\U(x0, y)|| < 1} is bounded, then dim, Y < dim, Z.
Let B (X , Y) denote the Banach space of all bounded linear maps from X into Y.
L
e m m a4 . Let ü
:X x B (X , Z) - > Z be o f the form U(x, V) = V(x).
I f X is isom orphic with Z, then the (B)-condition holds with respect to x.
C
o r o l l a r y1. Let р еШ , xne X , Vne B {X , X ), n = 1 ,2 , ... The con
vergence o f the series ]^ V n(xn) fo r every V. = {V n}e ЛМ[В {Х , X)) holds i f f x . — {xn}e Г М(Х).
We yet consider a special case of group algebra, namely L x — L 1(B) with usual norm ||a?|| — j \x(t)\dt and convolution (x*y) (t) = f x(t —
R R
— r )y {r )d r as ring multiplication. This is a commutative Banach algebra without unity. I t is well known that L x contains approximative unity, i. e. a sequence {yn} with the following property :
\\х * У п ~ х \\ 0 Y x e L l
(e. g. [1], P- 16).
The set {y e L x: ||ж0*у| К 1} is non-bounded for any fixed x 0 in L x.
To prove this, we suppose that for some x 0 Ф 0 the set {y : \\xQ*y\\
< 1} is bounded. Then according to Lemma 3 the functional ||y||0
= ||a?0*y|| is a norm in L x equivalent to the start norm ||-||. Applying the notion of approximative unity, we have \\x0* y n — x0\\ 0, consequently
\К*{Уп-Ут)\\ = \\Уп-Ут\\о-+Ъ if w, w -> oo. Since ||• ||0 ~ 11*11, hence there is an у in L x such that \\yn — y\\ -> 0. Thus we get
X * y n ~ > X — X * y Y i C e L j .
This is a contradiction since L x is an algebra without unity.
Nevertheless, by some supplementary hypothesis the statement of Theorem 2 for the bilinear series £ x n* y n, xn, yne L 1 remains also true.
For the purpose we consider the set В of all у in L x for which c f \y(t)\dt
R
< \jy(t)dt\, where c is a constants (0 < c < 1).
R
C o r o l l a r y 2. Let fo r every x. = {xn}e the series ]? x n* y n, where yne L x be L^convergent and let either (i) sn( y .) e B , n = 1 , 2 , ... i f у,€Ш0+ or (ii) rn(y.)e B , n = 0 ,1 , 2, ... i f уеШ ~ . Then y .e Г И(ВХ).
P ro o f. Let x0e В and ||a?0|| = 1. Since x0* y (which exists iff \x0 I* \y I exists) is in L x, by Fubini theorem
j (xx*y)(t)dt = J xQdt j у dt.
R R R
