A N N A L ES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)
Bor-Luh L
in(Iowa City) and Ivan S
in g er(Bucharest)
On conditional bases of l2
1. A sequence {xn} in an infinite dimensional Banach space E is said to be a basis of E if for any element x in E there exists a unique
00
sequence of scalars {an} such that x = aixi . A basis {a?n} is said to be
i — l
conditional (unconditional) if there exists (if there does not exist) a series oo
which is convergent but not unconditionally convergent. It is well
•i= l
known that {xn} is a basis (respectively, an unconditional basis) if and only if there exists a constant К > 1 (respectively, A unc > 1 ) such that
n n + m
i = l % = 1
for any scalars a17 .. ., an+m (respectively, such that
П П
{^) || ^ j ^гаг*^г|| ^ ^unc 11 Щ^г||
г= 1 г= 1
for any scalars cq, .. . , an, <5X, ..., ôn with lôj < 1 , ..., |<5J < 1). The least such constant C({xn}) = min A (respectively, Cuno({^}) = m inAunc) is called the constant (respectively, the unconditional constant) of the basis {xn}.
A basis {aqj is called bounded, if 0 < inf \\xn\\ < sup||£rj| < -{-oo.
П П
We recall that two bases {xnj and {yn} of a Banach space E are said to be equivalent if there exist constants Cx and C2 such that for any finite sequence of scalars ax, .. ., an
n n n
(3) < c 2 || 2 'aia?i||-
г= 1 г= 1 i = l
In the sequel, we shall make use of the following well-known theorem of Bari [2] and Gelfand [4] (see also [9]): In the space l2 all bounded un
conditional bases are equivalent.
136 B. L. L i n and I. S i n g e r
Although Hilbert spaces have “the best” geometric properties among all Banach spaces, the construction of conditional bases in separable Hilbert spaces appears to be more difficult than in the other concrete Banach spaces with unconditional bases. Since the existing construc
tions of conditional bases in the separable Hilbert space L2([ —тс, tc ]) (see Babenko [1], Gaposkin [3], Helson and Szegô [ 6 ]) lean heavily on analytic tools, it is natural to raise the problem of finding a simple geo
metric construction of conditional bases in l2 of the form
(4) con = ( w = l , 2 , . . . ) ,
i= 1
where {en} is the unit vector basis of l2. Moreover, with the Gram-Schmidt process we can obtain from any conditional basis {;xn} of l2 or А 2 ([—тс, тс]) an orthonormal basis {zn} of the form
n
Zn =
1
ф о II rH whence
n
г= l
A n) ф о {n = 1 , 2 , ...)
Since {;zn} is orthonormal, there exists a linear isometry и of l2 or L 2{{—тс, tu ]), respectively, onto l2, such that u(zn) — en (n = 1 , 2 , ...), whence the sequence
П
( 6 ) yn = «(*„) = № ф 0 (n = 1 , 2 , . . . )
i=l
is a conditional basis of l2. Thus, there arises the problem of finding explicitly a conditional basis of l2 of the form ( 6 ). (Let us observe that one cannot obtain the desired basis by applying the above procedure to the conditional bases of L2([—тс, тс]) constructed in the papers [1], [3], [ 6 ], since in these cases the a above involve integrals which are not computable.)
J. Lindenstrauss and A. Pelczynski have called to our attention that from a recent paper of McCarthy and Schwartz [ 8 ] follows the construction of a conditional basis of l2 of the form (4). Namely, from [ 8 ] one can obtain, for each щ a basis x^ , ..., x ^ of the №-dimensional Hilbert space l2 n such that 0 ({a?}n)}jL x) < I f < + oo, but Cfunc({®jn,}jL1) - > + o o as w- > + ° ° >
whence by [5], Theorem 2 or [10], Lemma 2, the sequence
(6) { o , 4 2> , o , . . .
•*• > • • • » 0 , 4 n\ о* • • ■ ■ ■ > {о? • • • j 4 4 o, • -•}i •
Л—1 71— 1
is a conditional basis of (Ji\ x l\ X ... X l2 nX . ..)гг. Hence by the natural canonical equivalence of (Jt\ X l\ x ... X l2 n X .. .)# with l2, we obtain a con
ditional basis of l2 of the form ( 1 ).
In the present note, still using the ideas of [ 8 ], we want to give somewhat more, namely, a conditional basis of l2 of the form (5) and a conditional basis of l2 of the form
00
(7) œ, = A"' *0 (» =1,2,...).
г
2. T
h e o r e m. The sequences {ccn} c- l2 and {yn} <= l2 defined by
00
(® ) 1 ^2n — 1H- ^ i —n + 1 ^2г ? ^2n ^2» • • • ) >
i = n n
( 0 ) V i n — X ^ 2 n — 1 ? У 2 n г + l ) ^ 2 г — 1 ~ Ь • • • ) ?
г= 1
{еи} denotes the unit vector basis of l2 and an > 0 (w = 1 , 2 , ...),
00 00
J j a * < + ° ° > 2 4 = o o ( e . y . , o w e c a n t a k e a n = l l ( n l o g n ) ) , a r e c o n d i t i o n a l
3 = 1 3 = 1
ôases of l2.
Proof. For any finite sequence of scalars j 8 1; we have
2 n n n с о n
P j X j f^2j — 1 ^ 2 j — l H~~ f t23 — 1 a i —з'+1^ 2г~ Ь ( ^ 2 j ^ 2 )
3 = 1 3= 1 3 = 1 г = з ) = 1
« n j
= ^ ^ 2 ) - 1 е 2 ) - 1 - \ ~ ^ ^ i^2fc— 1 a 3 - fc + l + /?2з) e 2j +
7 = 1 / = 1 & = 1
oo ?г
+
2 ( 2
/ ^ 2 f c - l a 3 -f e + l ) e 23?) ' = « ■ + 1 fc=l
whence
(10)
2n n n 7 oo n
jj /?y^-j| = £ \ ^ 2 j ~ l \ 2 j r ^ р 2 к - 1 а 7 - к + 1 + p 2 j I + I ^ 2 f c - l « 3 - f c + l| '
3 = 1 3 = 1 3 = 1 &=1 3 = и + 1 A=1
Since by the Holder inequality we have
OO n o o j n
£ I ^ & * - i a * - * + i | 2 < ( £ \ a k \ 2 ) £ \ p 2 i - x \ 2
3= w + 1 к = 1 з ' = п + 1 f c = 3 ‘ — n + 1 г = 1
oo n o o n
< 2 * Ü - 1 ) к -l 2 F ^ У i ai 2
3 = 2 г= 1 з*=1 i = i
138 B. L. L i n and I. S i n g e r
it follows that for any finite sequence of scalars , ..., §m with m > 2 n we have
7 = 1 &=1
^ I j°i) !/^ 2 ;-l|2+ j +A
7= 1 7 = 1 7=1
OO
Ш< ( г + 2 t â ) \ l 2 №
7 = 1 7 = 1
Similarly, for any finite sequence of scalars /Зх, ..., |Sm with m > 2n—l we obtain
2 n — 1 oo m
У fijXj ||2 < ( l + ^ jafj || JT (ijXj |j2.
7 = 1 7=1 ?= 1
Consequently, {xn} satisfies (1) with К ) / i + l « {xn} is a basis of l2. Let us also observe that
and hence
(
11
)11^271-1 \ Ы \ = 1 0 ® = 1 , 2 , . . . ) .
On the other hand, by ( 10 ) we have
n n n j oo n
||
^ 2 j - \ X 2 j - lj| = ^l^
27-ll2+ |
^ 2 k - l a i - k + lI + j
y 1, f t 2 k —l a j - k + l|
j= 1 3=1 3 = 1 k= 1
3=n
+ 1 k = ln n
?
^
У !I
$ 2 j —1 1I
@ 2 к - 1 a ) - k + lI 7
7
=
1 ?‘= 1 k =1whence, in particular, for /32?-_i = 1/Vn (j = 1 , ..., n) we obtain, taking П
into account that by our hypothesis j a?-j 2 -> oo as w -> oo,
7 = 1
it follows that {xn} is not equivalent to {en}, and hence, by the Bari-Gelfand
theorem mentioned above, {xn} is a conditional basis of E = l2.
Observe now that (xn, yn) is a biorthogonal system, since for all m, n = 1 , 2 , . . . , we obviously have
0 ^ 2 n i V ï m — \ ) i f 2 г г ? ^ 2 m — l ) ^ ? OQ
i . ^ 2 n — 1 ? У 2 т — \ ) ~ { ^ 2 n—1 “ H a i — n + 1 ^ 2 t ? ^ 2 m — l j ^ м ? г = п
m
i p ^ 2 n l У 2 т ) | ^ 2 г г ? ( ® m — г + l ) ^ 2 г — 1 " h ^ 2 m j ^ n m l г = 1
oo m
{ ^ 2 n — 1? У 2 т ) |^2и,—lH ~ n + 1 ^2г ? г+ l ) ^2г—1 H- ^2m j
г = » г г = 1
0 if m < n ,
г г + 1 ( ^ 2 ? г — 1 ? e 2 n - l ) H “ a m - n + l ( e 2 m l ^2m )
