Arch. Math., Vol. 43, 535-541 (1984) 0003-889 X/84/4306-0535 $ 2.90/0 �9 1984 Birkhfiuser Verlag, Basel

On a classification of sequences in Banach spaces

By

ANDRES REYES t

1. Pre l iminar ies and notations. Let B denote a separable Banach space and B* its dual, ~9 ~ = (a,),~ N a general sequence in B and [ ] the closed l inear span, ~r 1 = {S c N; S is finite}, ~r 2 = {S _~ N; S is cofinite} an d 0- 3 = {S c N; S is infinite with infinite complement} .

1.1. D e f i n i t i o n s. The Kernel of 6 e is

g ( ~ ) = (~ [an, an+ 1 . . . . ] , n~N

or equivalent ly

K ( ~ ) = (~ [ak; k ~ S], S~a2

and its strict Kernel*) is

K~(5 a ) = ~ [ a k ; k ~ S ] . S~a3

Call

M ~ = {n~ N; a , ,~[a. , ;m~ N \ {n}]}.

The sequence ~ = (a,),~N is called minimal (or topological ly free) if M ~ = 0, of absorbent kernel if [ak; k ~ M~] c_ K(SQ an d of absorbent strict kernel if [ak; k ~ M~] ~ K s ( ~ ).

O b s e r v a t i o n. I t is easy to prove that 6 e is minimal if and only if

[ak; k ~ S] n [ar; r ~ T] = [ah; h e S ~ T],

for any S ~ al , and T~ a 2. In [2] it is proved that 6 e = (a.),~u has absorbent kernel if and only if

[ak; k ~ S] ~ Jar; r ~ T] = [ah; h e S c~ T],

for any S, T~ a 2 .

*) In [5], Terenzi uses the word "nucleus" for strict Kernel.

5 3 6 A. REYES ARCH. MATH.

O n this line, a result of this paper is:

5 p = (a.).~ N has absorbent strict kernel i f and only i f

[ak; k ~ S] c~ [G; r e T ] = [ah; h e S c~ T],

for any S ~ a 2 and T~ a 3. If 5 ~ = (a,)n~ N is a complete min ima l sequence in B and 5 ~* ----= (an)he N * C B* is the

conjugate sequence of 5 P, for any subset S _ N call

W s = [ a k ; k e S ] , and

W~" = k(~S Ker a~'.

We assume that W 0 (0 the empty set) is the nul l subspace and W* = B. Observe that W~* = K(Se), and Ws ~ Ws* for every S ~_ N.

1.2. D e f i n i t i o n. ~ is called strong M-basis if

w~= w~,

for any subset S _~ N. If b ~ is no t complete, 50 is strong M-basic if 5 ~ is a s t rong M-bas is of WN. F o r further studies on s t rong M-basis theory, see [4].

1.3. Lemma. For every sequence 5P = (a,),~ N in B, and all x ~ [an; n ~ N] there exist two subsets S, T ~ a 3 such that

X ~ X 1 "~- X2~

with x l ~ [ak; k E S] and x2 ~ [at; r 6 T]. (See [6]).

This result, due to P. Terenzi, closes an open p rob lem posed by P lans and Reyes [2] and gives the fol lowing intr insical character iza t ion for s t rong M-bas ic sequences:

A sequence 5 r = (a,),~ N in B is a strong M-basic sequence i f and only i f

[ak; k ~ S] c~ [at; r ~ T] = [ah; h ~ S c~ T],

for any S, T 6 a 3. Final ly, in order to ob t a in the m a i n result, concern ing the na tu re of those sequences

5 ~ = (an).~ N in B verifying

[ak; k ~ S] n [at; r e T] = [ah; h ~ S ~ T],

for any S e a 1 an d T e a 3, we use the following

1.4. D e f i n i t i o n. If St = (a,),~N is a min ima l sequence in B, a vector u in [a,; n ~ N] is in an unitarian position (u.p.) with respect to a subsystem (ak)k~S of J if U e [ak; k ~ S] and u ~/[at; r e S - {k}], for any k e S.

Vol. 43, 1984 Sequences in Banach spaces 537

O b s e r v a t i o n s. If u ~ WT* is in an u.p. with respect to a subsystem (ak)kES, then S _ T. Therefore, the subsystem for which a fixed vector u is in an u.p. is unique. Moreover , if Sf = (a,),~ n is a m i n im a l sequence such that every vector u ~ [an; n ~ N]\{0} is in an u.p., then 5 a is s t rong M-bas ic and conversely. (See [3]).

1.5. D e f i n i t i o n. A general sequence 5 e = (a,),~N in B is p-minimal for some p e N if there exist na tu ra l n u m b e r s k 1 . . . . . kp such that (a.),~N\ (%)~P= 1 is minimal , and p is the smallest n u m b e r with this property.

We assume that 0 -min imal sequences are min ima l sequences.

1.6. L e m m a (see [1]). 6 P = (a.).~ N is not p-minimal for any p i f and only i f there exists

S ~ a 3 such that

[a k; k e S] = [a,; n ~ N].

The m a i n result ob ta ined is the following: A sequence 5e = (a,),~ N in B verifies

[ak; k e S] c~ [a,; r s T] = [ah; h ~ S c~ r] ,

for any S E a~ and T e 0-3 i f and only i f 5 a is a p-minimal sequence, moreover i f 5 ~ = (bk)f= 1 U (bj)j > p is a corresponding decomposition o f 5", then (bk)~= 1 is linearly inde- pendent and every vector b ~ [ b i , . . . , bp]\{0} is in an u.p. with respect to a cofinite sub- system o f (bj)~ > p.

2. Characterization of U WT" If 5 r = (a.),~ N is a complete min ima l sequence in B, T~o" 3

then it is s t ra ightforward to prove that Ws* = W s for any S ~ 0-2. It is no t true, in general, for S E 0-1 u 0-3. If S ~ a l , then W * = W s �9 K(5r We have the following:

2.1. Proposit ion. Given S ~ 0.1 u a3, there exists T~ 0- 3 such that W~ ~_ W r.

P r 0 0 f. It is clear if W* = l/Vs. Assume then that Ws* ~ Ws, S ~ 0-3, and call S = {10,; n ~ N}. We can extend (ap.),~n with vectors in W ~ ' \ W s to ob t a in a complete sequence in Ws*

(ap.),~N u (bl , b2 . . . . ).

There exists R 1 = {11,... , lrl } ~ 0-1 such that

bl I'1 - Z 2~l)ak < 1. k =11

Let t 1 ~ N \ S a n d q > Ir~. O n accoun t of bl ~ Ws*, ba ~ [a,; n ~ N \ { t l } ] and therefore exists R 2 = {m 1 . . . . , m~2 } c N \ { q } such that

m r 2

b 1 - ~ 2(k2)ak < 1/2. k = m l

538 A. REdS ARCH. MATH.

Let t 2 �9 N \ S and t 2 > max{t1 , m~}. Since b2 �9 Ws*, b2 �9 [a.; n �9 N \ { t l , t2}]. There exists thus R 3 = {n~ . . . . . n~} ~ N \ { t ~ , t2} such that

n r 3

b 2 - ~ ~,~3)a k < 1/2. k = n l

Let ta � 9 and t a > max{t~ , t2, n~}. As before, we have a R 4 = (Pl . . . . . p~} c

N \ { t l , t2, t3} for which

Pr4 b l - Z 2(k 4) ak < 1 / 3 .

k = p l

Take now t4 e N \ S with t 4 > m a x { t l , t2, ta ,p~} . There exists R 5 = {ql . . . . . q~) ~

N \ { t l , t 2, t 3, t4} such that

b 2 qr5 -- Y~ 2(kS) ak < 1/3. k = q l

Fo l lowing on, we wou ld have t5 �9 N \ S and R 6 c N \ {t 1 . . . . . ts} for which

b3 - k ~ 2(k6) ak < 1/3.

I te ra te indefini tely this process to ob ta in {t,; n �9 N}.

Le t T = N \ { t , ; n �9 N} . T h e n a p e Wr and, by const ruct ion , b , e Wr for every n �9 N. So

that IV* = [ap.; b~, b 2 . . . . ] ~ Wr. Final ly, if S e a l , since S ~ R for some R �9 a3, the

result fol lows obviously. []

2.2. Proposition.

{x �9 B; x is not in an u.p. with respect to any cofinite subsystem o f S P} = 0 W*. TE~3

P r o o f . F o r any x � 9 denote S x = { n � 9 Assume x0~ WT* for any

T e a a . Since x e W~ it fol lows that Sx e a2. We have

(i) x �9 w~* = ws~, (ii) x di Wsx\lkl for any k e S x. Otherwise, if x �9 Wsx\{k} for some k �9 Sx, since

x = a* (x) ak + Xk, Xk e WN\{k}, then

a* (x) a k = x -- x k e WN\(k }

and a* (x) = 0, which cont radic ts k e S~.

Therefore , x is in an u.p. wi th respect to the cofinite subsys tem (ak)k~Sx. Conversely , let x be in an u.p. with respect to the cofinite subsys tem (ak)k~S, and

suppose tha t x �9 W* for some T e a 3. T h e n S __ T, which is no t possible because of S �9 cr 2 and T e a 3 . Hence x ~ WT for any

T e a a and this concludes the proof. []

As a consequence of 2.1. and 2.2. we have

Vol. 43, 1984 Sequences in Banach spaces 539

2.3. Corollary. {x ~ B; x is not in an u.p. with respect to any cofinite subsystem o f Sf} = U WT" []

T ~ a 3

3. Characterization of the sequences (a.).~ N in B such that

[ak; k ~ S] c~ [ar; r ~ T] = [ah; h ~ S c~ T]

for any S ~ a z and T~ a 3 (1). Let 5 # = (an)n~ s be a sequence in B.

3.1. Theorem. 5 p verifies (1) if and only i f it has absorbent strict kernel.

P r o o f . Suppose tha t Ks(5 e) is no t absorbent . W e can assume wi thou t loss of genera l -

i ty tha t a s ~ [a2, a 3 . . . . ] and a s ~ [at; r s T] for some T~ a 3. Then

[as, ar; r ff T] ~ [a2, a 3 . . . . ] ~ [ar; r ff T],

and therefore (1) does no t hold. Converse ly , suppose tha t Ks(5 e) is a b s o r b e n t bu t (1) fails. There will exist S e 0.2 and

T c a a such tha t

[ak; k e S] n [ a , ; r e T] ~ [ah; h e S ~ T].

Call S c~ T = { p . ; n E N } E 0 . 3 ,

T~ S = {rD . . . , rh} ,

and choose x e [ % , . . . , a~; ap., n e N] c~ [ak; k e S] and x # [a , . ; n e NI. Then

x = 21a~ + . . . + 2ha~ + X', X ' e [ap.; n e N],

wi th 2~ =t= 0 for some i e (1, . . . , h}. So tha t

a r e [ak, k E S; a . . . . . . , %_~, a . . . . , . . . . % ; ap., n e N] ~ [am; m e N\{r l} ] .

Since K~(50 is absorben t , it fol lows

a,, e K~(5 ~) c [ap.; n e N],

which is a con t rad ic t ion . []

E x a m p 1 e s. Obvious ly , every m i n i m a l or overf i l l ing sequence 5 p (i.e. [5 a] = [Se'], for any subsequence ~ ' of 5 0 has a b s o r b e n t s tr ict kernel.

N o n t r i v i a l e x a m p l e . Let ~ = ( X n ) n ~ N be a min ima l sequence in B with K~(SP1) =t = {0}, and 5P2 = (Y,),~u a comple t e overfi l l ing sequence in K~(~ ) . Then the j o in of 5e~ and SP z has a b s o r b e n t str ict kernel and it is ne i ther min imal , no r overfill ing.

4. Nature of the sequence (a,).~ N in B such that

[ak; k ~ S ] n [ar; r ~ T] = [ah; h ~ S ~ T],

for any S ~ 0.2 and T e 0 3 (2). Fi r s t we s tudy an example . Let 5 e = (a,),,~N be a comple te

540 A. REYES ARCH. MATH.

min ima l sequence in B and u ~ B in an u.p. wi th respect to 50 (such a vec tor exists since U K e r a* :t= B). The sequence {u} u (a,),~ N has the p r o p e r t y tha t all its subsequences,

h e n

whose defining set of indexes is in r are minimal . In fact this p r o p e r t y character izes the sequences for which (2) holds.

4.1. Proposi t ion. For a sequence 50 = (a,),~N, the fol lowing are equivalent:

(a) Every subsequence o f 5~, whose defining set o f indexes is in a3, is minimal. (b) 50 verifies (2). (c) aiOi[a j ; j eR] , for any R e a 3 and iOiR.

P r o o f. I t is obv ious tha t (b) implies (c) and (c) implies (a). Let us p rove tha t (a) implies (b). F o r any S e a i and T e a3, take 5~ = (aj)j~s ~ r . By hypothes is , 5~i is min ima l and, by the obse rva t ion m a d e after 1.1., it follows (2). []

W e are r eady to p rove our ma in result.

4.2. Theorem. A sequence 50 verifies (2 ) / f and only i f 50 admits a rearrangement such that

b v 50= ( k)k= l U (bj)j>p, being

i) (b i . . . . . bp) linearly independent and b i e [bj;j > p], for every i E {1 . . . . . p}, ii) (bjb> p minimal,

iii) every vector b ~ [b i . . . . . by] \{0 } is in an u.p. with respect to a cofinite subsystem of (b,)j>p.

P r o o f . By 1.6. if 5 ~ verifies (2) then 50 verifies i) and ii). C o n d i t i o n iii) fol lows by 2.3. and 4.1.

Converse ly , given S ~ a i and T e a3 disjoint , wri te

S i = S c~ {1 . . . . ,p}, T i = r ~ {1 . . . . . p},

S 2 = S - S i and T 2 = T - T 1 .

F o r x e [b A s e S] c~ [bt; t ~ 7],

x = x l + x2, x l e [bi; i ~ Si] , x2 e [bj;j e $2] and

x = Yl + Y2, Yi ~ [b,; i ~ TI], Y2 e [ b / j e T2I. Thus

Z = X 1 - - Y l = Y 2 - - X 2 ~" [bl . . . . . bp] c~ [bl; l e R],

wi th R e a a. I t fol lows by cond i t ion iii) tha t z = 0. Then x i = Yl and

x2 = Y2 e [bA i e $2] c~ [bj;j ~ TEl i~ {0}.

W e conc lude

x = x 1 = Yl e [ b , ; i e S 1 ] c~ [ b j ; j e T1] ~ {0}. []

Vol. 43, 1984 Sequences m Banach spaces 541

5. We have the fol lowing scheme for sequences in B: 1) [ak; k ~ S] c~ Jar; r ~ T] = [ah; h ~ S n T], for any S, T~ a I if and only if 5O is linearly

independent. 2) [ak; k ~ S] c~

bent kernel. 3) [ak; k e S] c~

M-basic. 4) [ak; k ~ S] c~

minimal. 5) [ak; k ~ S] ~ [at; r e T] = [ah; h e S c~ T], for any S e al and T e a 3 if and only if 50 is

a p-minimal sequence accomplishing the conditions i ) - i i i ) of the 4.2. Theorem. 6) [ak; k ~ S] c~ [a,; r ~ T] = [ah; h ~ S c~ T], for any S ~ a2 and T~ a 3 if and only if 5O

has absorbent strict kernel.

[at; r ~ T] = [ah; h E S c~ T], for any S, T~ 0 2 if and only if 5O has absor-

[G; r ~ T] = [ah; h ~ S n T], for any S, T e a3 if and only if 5 ~ is strong

Jar; r e T] = [ah; h E S c~ T], for any S ~ al and T~ (r 2 if and only if 50 is

References

[1] V. D. MIL'MAN, Some properties of unconditional bases. Dokl. Akad. Nauk SSSR 162, 269-272 (1965).

[2] A. PLANS and A. REYES, On the geometry of sequences in Banach spaces. Arch. Math. 40, 452-458 (1983).

[3] A. REY~S, A geometrical characterization of strong M-bases. Rev. Roumaine Math. Pure Appl. (to appear).

[4] I. SINGER, Bases in Banach spaces II. Berlin-Heidelberg-New York 1981. [5] P. TERENZI, Biorthogonal systems m Banach spaces. Riv. Mat. Univ. Parma 4, 165-204 (1978). [6] P. TERENZI, Representation of the space spanned by a sequence in a Banach space. To appear.

Eingegangen am 15. 11. 1983")

Korrespondenzadresse:

Antonio Plans Departamento de Geometria y Topologia Facultad de Ciencias Zaragoza, Spanien

*) Eine Neufassung ging am 23.3. 1984 ein.