XU On bilinear series in Banach spaces, I

ANNALES SOCIETATIS MATIIEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1967)

H. Sampławski (Gdańsk)

On bilinear series in Banach spaces, I

1. Introduction. The purpose of this paper is to inwestigate the convergence of bilinear series of the form

CO

(1) ^ U ( x n, y n)

n= l

where J 7 : l x l ^ ^ i s a (bounded) bilinear operator from the product of Banach spaces X , Y into a Banach space Z. By the convergence we shall always mean the norm convergence in the space in question. In particular, if X is the set E of real numbers, Z — Y and U (x, у ) = xy, then (1) is just the series

00

(2) xnyn.

n= 1

If X — Y and xn = у n, (1) is a series ]?V(xn), where V : X ^>Z is a ho­

mogeneous quadratic operator.

The author is indebted to Professors Władysław Orlicz, Eustachy Tarnawski, and Zbigniew Semadeni for helpful criticism and a large number of suggestions contributed during the preparation of this paper.

2. Comparison functions. The symbols 9}^+, 9Л0, and

XU

Definition. Let {xn} с X . A sequence у = {y(n)} is said to be a comparison function for the series ]?xn provided that ye Xfl and (3) sup —y-y < cx?, i.e. K ll = 0(y(n)),

n у \П)

where an — + ... -\-xn, and is said to be a comparison function of posi­

tive order, of order 0, of negative order, provided that у eXfl+, у eXfc0, у eXft~, respectively.

10 — Prace Matematyczne XI (1967)

Let us notice that every series £ x n has at least one comparison function of positive order, e.g.,

( IN I + ••• + 1 Ы if ^ I M = 00, fx{n) =

\ n if 2j 1Ы < °°-

A series f ? x n has a comparison function of negative -order if and only if limcrn = 0, and has a comparison function of order 0 if and only if it is hounded. Le t us also notice that if x n Ф 0 for n — 1, 2, ..., then there e xists a comparison function of negative order which is not a comparison function for the series £ x n . I f ... is an orthonormal system in a H ilb e rt space, then y ( n) = n is a comparison function for ]?(pn .

I f julf y 2e cffl, let [лх -<§ ju2 mean that jux{n) — 0{/и2{п))-, y x, y 2 are called equivalent (written y x~ y 2) if y x - 3 y 2 and y 2 y x.

3. Criteria of convergence.

Lemma 1. Let w ^Ш°. I f the series

OO

(4) ^ y { n ) A y n, where A y n = y n — y n+x, ii—i

is weakly unconditionally convergent, i.e., i f

OO

(5) ^ y ( n ) \ f ( A y n)\ < 00

n= 1

for every f i n У*, then the sequence y{n)\\yn+x\\ is bounded.

P ro o f. I f and/e Y*, then by Kronecker’s theorem ([6], p. 131), lim (|/(y1- y 2) l+ ••• +№(Уп—Уп+i)l)^(^) = 0

n—>00

and hence \ f ( y i ) —f ( y n+i)\ft(n) 0. Since y ( n ) j 0, th is yields f ( y n+i)y(n) ->

-> 0, which means that y { n ) y n+x tends weakly to 0 and y (n) \\y n+x\\ — 0 (1 ).

blow, if jue^ft 0, i.e., y ( n ) = c > 0, then £ \ f ( A y n)\ < 0 0. Consequently, {yx—y n} is weakly convergent and therefore both {yx—yn} and {yn} are bounded.

Lemma 2. Suppose that y e cAIl~ and

OO

(6) ^ y{n)\ \Ayn\\ < oo.

n—1 Then lim y{n)\\yn+x\\ = 0.

n—>00

The proof is analogous to that of Lemma 1.

Lemma 3. Let y e c^Sl+. I f series (4) is wealdy unconditionally conver­

gent and {yn} is weakly convergent to 0, then {/л{п)уп} is weakly conver­

gent to 0.

Proof. If f e Y * and (4) is weakly unconditionally convergent, then there exists an N snch that

г- i y W \ f ( y k-y i)\ = Ayn)

n = k

l- l l- l

< \f(Ayn)\ < £ P(n ) \ f(A y n)\ < S

n = k n= k

for l > к > N. Keeping к fixed and passing to the limit with l -> oo we infer that y{k)\f(yk)\ < e for к > N.

Lemma 4. I f jue<Ntl+, yn -* 0, and (6) holds, then y(n)yn -> 0.

The proof is analogous to that of Lemma 3.

Theorem 1. Let ooneX, уп*¥, let y,(n) be a comparison function for the series ^ fx n, let yn -> 0, and let

OO

^ y ( n ) \ \ A y n\ \ < ° ° .

П— 1

Then series (1) is convergent in Z. Moreover,

. OO oo

TJ{xn, yn) = TJ(an, Ayn)

n= n=l

and

oo oo

(7) ||J^ U(xn, yn)\\ 2r{n)\\A yn\\,

W = 1 П—1

' K il

where К = ||Z7||sup---and an = x x-\- ... + a?n.

/u(n)

Proof. Applying Abel’s formula ([1], p. 68) to the sequence { 8 k }

of partial sums of (1) we get

* -1

8k ~ L(on, Ayn) C7((Tfc, Ук).

n= 1

The series ^ U ( a n, Ayn) is absolutely convergent because

oo oo

A 11ЕГО., Ayn)W <liC_yV(»)|Mj/„||.

W=1 n= 1

Since \\U(Ok, у к) || < Кр(Тс)\\ук\\, in virtue of Lemmas 4 and 2, the sequence 8k is convergent. Moreover

OO 00 o o

= \\^ T J ( a n,A y n)\\ < К 2 у(п)\\Ауп\\.

тг= 1 п —l п = 1

E e m a r k 1. If р е (2Я~, then the assumption yn -> 0 in Theorem 1 is superflous.

Theorem 2. Suppose that xneX, yn*Y, pzXf t T, that the series ^ x n is convergent to a?0,

ll^o — °n\\ = 0(p{n))

where an = aq-f- ... +a?n, and that (6) holds. Then (1) is convergent and for every m

J

n = m

< \\U|| supIHo-

p{n) p ( m —1) + J ? у{п)\\Луп\\

Proof. Denote r0 = x0, rn = x0 — on (n = 1, 2, ...). Then ||гте|| -> 0.

Applying Abel’s transformation we get

p p —i

(8) hi (xn , уn) — ^ hJ(rn , yn+1 yn) A U (rm_u Уш) U (Гр , yp) .

n = m n = m

The first term on the right-hand side is convergent as p -> oo because

p—i

2 U{rn, Ayn)|| < || Z7|| sup П

INI

p( n)

P—1

^ y(n)\\Ayn\\.

n = m

Moreover, \\U(rp, yp)\\ < || U\\y(p) ||yp||sup\\rn\\/p(n) and therefore, by Lem- П

ma 2, || U{rp, yp)\\ -> 0 as p of (8) we also have

p

U (xn, yr

П — ГП

< II ?7|| sup_Ы p(n)

oo. Thus, series (1) is convergent. In virtue

p— i

[ ^ y ( n ) \ \ A y n\\+iu(m-l)\\ym\\] + \\U{rp,y p)\\.

n=m

Passing to the limit as p oo we get the desired conclusion.

Theorem 3. Suppose that series (4) is wealdy unconditionally conver­

gent. Suppose also that either (i) peXfl°~ or (ii) ресШ + and {yn} tends wealdy to 0. Furthermore, suppose that {xn} is a sequence of numbers such that lim an/p(n) = 0, where on = aq + ... +a?n. Then series (2) is conver­

gent in Y and for every f in Y* the inequality OO

n = 1

< sup V p (n) I/ ( Ayr, n p{n)sLJ

holds.

k- 1

Proof. Denote ak — £ <Jn Ayn and j3k = okyk. Abel’s formnla gives

<*k+Pk = sk = £ хпУп• By Lemmas 1 and 3, the sequence y(k)\\yk\\ is bounded and therefore

W k\\ m \

[л(к)в(Щ Ы \ ⁰.

Since £ у ( п ) Ayn is weakly unconditionally convergent, for every sequence {tn} of numbers convergent to 0 the series ]?tnp(n) Ayn is convergent in Y ; in particular, setting tn = ап/[л(п) we infer that the series anAyn is convergent. Consequently, sk = akArfik is the sum of two convergent sequences. If f e Y*, then

к

I^ xnf ( y n)

n= 1

к

< l / K ) l + i/(A )l < Y ~ l * ( n ) l f ( A y n)\+o{l);

A ' /“<») passing to the limit we get the desired inequality.

4. Examples and applications.

Example 1. Let /иеЧЯ, ]?[11в(п )12 < °°> let (4) be weakly uncondi­

tionally convergent, and let {yn} tend weakly to 0. Then the series £ y nrn(t), where rn denotes the n-th Rademacher function ([1], p. 51-52), is convergent (in norm) almost everywhere on <0,1).

Proof. By Bademacher’s theorem ([1], p. 52) the series

I

rn(t) y(n)

is convergent a.e. on <0,1>. Applying Kronecker’s theorem ([6], p. 131) we infer that

• • • A-i'nil) _^ q

fx(n) a.e. on <0,1> and we apply Theorem 3.

Example 2. Suppose that

< OO,

(4) is weakly unconditionally convergent and {yn} tends weakly to 0. Then for every orthonormal system {q>n} in А2(0,1) the series ^ у пУп(1) is con­

vergent a.e. on <0, 1>.

This follows from the Menchoff-Rademacher theorem ([1], p. 76) in a way analogous to that above.

Example 3. I f {yn} is a sequence of numbers tending to 0 and

oo

J ? n llP\Ayn\ < oo n=^l

where 1 < p < oo, then

OO OO

(9) ( IУп\Р)11Р < nllvI AynI.

n=1 n=1

Proof. Let X = Z == lp, Y = B, U(x, y) — yx and let xn be the n t h unit vector (0, . . . , 1, 0, ...). Since

\\Gn\\ip — ^1/P and || P’11 = 1,

and the series £ y nXn is ^-convergent to {yx, y 2, ...), we may apply The­

orem 1 with К = 1.

R e ma rk 2. Inequality (9) may be compared with Example 1.

Indeed, suppose that {yn} tends weakly to 0 and

OO

£ V n \ f { A y n)\ < oo.

n=\

Then, by (9), ^ | / (2/и)|2 < oo and therefore for every / in У* the series 2,f{Vn) rn(t) is convergent a.e. on <0,1>. This conclusion is weaker than that of Example 1, but the assumptions are also weaker (e.g. consider /.i{n) — Vn).

Example 4. There is no orthonormal system {<pn} in _L2( 0 , 1) such that each rpn is bounded and that

| | < P l + • • • -ftPnWLn — P ( ^ 0 ( ^ ) ) j

wherep0{n) ~ n i~E (log n f , 0 < e < 1/2, Tc > 0.

Indeed, if such a system existed, then for every sequence {an} ten­

ding to 0 and such that f£[t0{n)\Aan\ < oo the series ]?ап(рп would be uniformly convergent a.e. (in virtue of Theorem 1). This, however, is not true for an = l j] /n whereas

Y *<»>i f - - Х Л = V < у ( ^ < CO.

Example 5. Suppose that a > 1 and

an~\-^n+i + ••• = 0((logw) 2a),

then for every orthonormal system {грп} in L2(0,1) the series ^ a n<pn(t) is convergent a.e. for any rearrangement of its terms. (This is a special case of a theorem of Leindler [7], p. 113, Theorem 1; cf. also [2], p. 345).

Indeed, denote /л(п) = (\ogn) yn — (\ogn){k+1)lk, where к is an integer and к > 1, and

\ log (n -1—1) 1 M = sup[ ’ ; n = 2,3, ... .

log n

Applying the elementary identity ak+l —bk+1 = {а—Ь)]?ак~гЬг we get

\Ayn\

= [(lOg(» + l))1,<:— (lOgM)1'*] У (log(» + l))(i *,,l'(logn)‘

i— 0

< (jfc+l)log(w+l) [(log(w-j-l))1/fc — (logw)1/fc]

,, , log(w+l) — \ogn

= (fc+l)log(w+l) fcZI---:— --- E (log(w+l))(fc-1- t)/fc(logw)ł/fc

i = 0

ijk

+ (fc + l ) l og( w+l ) l og (l + l/ w)

Tc(logn)

, 1 \ (logw)1/fc + (1 -|---- \ M —— ;

к I n

therefore, if a > 1 + 1 /к, then E Iй (n ) \ДУп\ < °°- In virtue of the assump-

, n=2

tion above,

oo

|| Л «т^тЦх.г (logw)a = 0(1);

m=n- f 1

hence, by Theorem 2, the series Е апУп<Рп is convergent in L %. Therefore E a 2n{logn)2+2lk < oo and our assertion follows from a theorem of Orlicz ([1], p. 108).

Example 6. (Plessner’s theorem, cf. [2], p. 341). I f ^a > \ ^{a n d}

£n + £>«+i + ... — 0 ( ( l o g n ) 2a),

w h e r e g l = a2n+&«, t h e n ^ ( a ncosw4+&risnm£) i s c o n v e r g e n t a . e .

Indeed, denoting ^{/ u ( n )} = (logw)~“, ^{y n} = (logw)1/2 and applying the Kolmogorov-Selivestrov-Plessner theorem ([2], p. 332), we get the con­

clusion in a way similar to that used in the preceding proof.

5. The space Л* for

ye

/иеУЛ0^

A

OO

№11 = ^ У (n ) МЛг| <

n= 1

oo.

It is clear that Л^ is a Banach space. We shall show that the unit vectors e(1), e(2), ... form a basis in it. Indeed, if X — {Xn} e A then (applying Theorem 1 to X = Z = Y = B, xn = e{n), yn — Xn) we infer that the series is convergent in a straightforward computation shows that this series is convergent to X and that the expansion is unique.

Theorem 4. The general form of a linear functional on Л^ is

(io) m =

П—1

where qn are numbers and

imi l(?i + • • • + 941 . ' [(/[I = s u p --- — --- < oo.

n [A (^)

Proof. If feA * , denote qn = /(e (n)). Since {e(n)} is a basis, f is of the form (10). Moreover,

lffi+ ... +ff*I = I/(+> + ... + + l))| < \\f\\y(n)

and hence s u p l ^ - f . . . +qn\[y(^) ^\\f\\- The converse inequality fol­

lows from Theorem 1.

Lemma 5. A sequence у in Xft is a comparison function for a series ]>fxn if and only if for every f in X *

1/(^1 + • • • f-Xn) I sup--- --- < OO .

n [Л{п)

This follows from [5], p. 255.

Lemma 6. Let qn be a sequence of numbers. Then £ X nqn is convergent for every X = {Aw} in if and only if у is a comparison function for ^fqn.

Proof. The sufficiency follows from Theorem 1. In order to prove the necessity let us consider the sequence {fm} of linear functionals on Лц defined as

fm W = where q\

Then

ll/mll = SUp

к

k r + . - . + qj

!i(Tc)

(m ) i

(/) _ qn for n < m 0 for n > m

= sup ••• +9+

By assumption, for every X in Л„ the sequence {fm{X)} is convergent as m - > o o . Hence (see [5], p. 229, Theorem 1)

sup||/m m

l^i+ ••• +(Ы su p ---

к ju{k) ^{< oo,} and the desired conclusion follows.

Theorem 5. I n order that ^]Xnxn be convergent for every X in Лй it is necessary and sufficient that у(п) be a comparison function for ^ fx n.

Proof. The sufficiency follows from. Theorem 1, and we shall show the necessity. Let ns notice that if for every X = {Xn} in Л^ the series 2\Xnxn is convergent in X , then the series £ X nf ( x n), where / is any element of X*, is convergent as well. Thus, the desired conclusion is a consequence of Lemmas 5 and 6.

C^orollary 1. I f the series ]?Xnxn is weakly convergent for every X in Л then it is strongly convergent for every X in Л

Proof. Since the series ]?Xnf ( x n) is convergent for every X in Л*

and every / in X *, p is a comparison function for the series f>fxn (by Lemmas 5 and 6); therefore our assertion follows from Theorem 1.

Now, let X be a Banach space with a basis {уп}. If XeA^, denote

oo

TJ(X, x ) Xntn(x)rjn

n = l

where {tn} с X* is the system biorthogonal to {gn}.

Pr o po sit io n. The operator U : Лх x X -> X is bilinear (here Лг means Л{1)1>>>.}).

Proof. For every x in X the series tn(x)rjn has a comparison func­

tion of order 0 (p(n) = 1 for n — 1,2, ...). Thus, by Theorem 1, U is well defined on Л г х Х and

к oo

\\U{X, x)\\ < s u p ^ t n(x)r]Ąj?\AXn\.

k n = 1 n =1

Since there exist positive constants ct and c2 such that

к

Cl IN < sup

II

* n =1

([8], p. 151-154), we get \\U(X, a>)\\ < с2р||||<в||-

6. The space Л

^

OO

Pll =

<

n= 1

In this ease {Xn} need not tend to 0, but again Л^ is a separable Banach space.

Th eo r em 6. The general form of a linear functional on Ли is (10) with ll/ll = sup

П

1 ^ + 1 ~\~Цп+2 + • • - I

p{n) < OO .

Proof. We begin as in the proof of Theorem 4 and estimate

|gVn + ••• +<Zml — I/(e(w+1)+ ... + е (ж))|

< ll/ll ||e<n+1>+ ... + e (w)|| = \\f\\(y(n)+y(m)).

If m -> oo, then ^ ( т ) - > 0 and therefore 1<7п+1+9»Ч-2 +

sup < 11/11 •

The converse inequality follows from Theorem 2.

Theorem 7. Letf The series £ X nxn is convergent for every A in Лц if and only if

snpП

\ \ X n + l Jr 0Cn + 2 JT • • •

ju(n) < oo.

The proof is similar to that of Theorem 5. We can also conclude that if ^ X nxn is weakly convergent for each Ain Л^, then it is strongly convergent as well.

7. The Haar series of Holder functions. Let

co(h)

tinuous decreasing function defined for h > 0 and such that cofh) -» 0 as h 0, and let

limsup--- < oo.h о (o(h)

A function / is said to satisfy condition (a generalized Holder condi­

tion) if \f ( t- fh )—f (t)\ = O[(o(hj) as h 0.

The Haar function (cf. [1], p. 48) are defined as folows:

and

zi0)«) = i

^2” for ( * - -l)/2n < t < ( 2 k - l ) / 2 n x P m = ¹ q ISI c for (2k-- l ) l 2 n+1 < t < Jc/2n,

0 elsewhere in <0, 1>,

where n = 0,1, ...; к = 1, 2, . . . , 2n. This double sequence is rearranged into a single one by setting ^(00)(«) = X\ and x2n+k = Xn] for n = 0, 1, 2, . . . ; к = 1, 2, ..., 2n. The series

oo 1

Cn(f) Xn(t), where cn(f) = f f{t)xn(t)dt,

n=l о

( И )

is the Haar series for /. It is known that for every continuous function / on <0, 1>, series (11) is uniformly convergent to / (cf. [1], p. 49), and /

satisfies condition Нш if and only if П

! ) /- y ^ ( / ) » | c<0,1} = o ( ® ( i » )

k = l

(cf. Golubov [4], p. 1274-1276). Combining this with Theorem 7 we get Theorem 8. Let f e C< 0 ,1 ) . In order that f satisfy condition Н ш it is necessary and sufficient that the series

OO

*nCn(f) Xn(t) n=1

be uniformly convergent for every 1 in A/e, where у (n) = co(l [n).

Combining a theorem of Ciesielski ([3], p. 156) we get a similar the-

OO

orem for the Franklin series S Oi f:

n=l

Th eorem 9. A continuous function f on <0,1> satisfies the Holder condition with an exponent a (0 < a < 1) if and only if the series

CO

] ? K b n{f)<Pn{t) n= ¹

is uniformly convergent for every X in А where y(n) = n~a.

Corollary 2. I f f satisfies Holder condition with an exponent a, then

j| ^ n fibn(f)tpn = 0

m where 0 < P < a .

n=m ' '

Corollary 3. I f f e C ( 0 , 1>, then £ n ~ abn(f)q)n satisfies Holder's con­

dition with exponent a since

CO

L , n - aK{J)Vn || = 0 ( l j m a).

Indeed, the Franklin system is a basis for <7<0,1> and therefore for every / in (7<0,1> there exists a in 9Л such that |K(/)|| < K pf (n).

Since X = {r—“} e Apj, Theorem 2 yields the estimation

OO

n~abn{f)(pn

n —m

< K m p p f (n) П

1

( » + ! ) “

)]

and we may apply Theorem 9.

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