1. Projections in locally bounded spaces. Introduction. Projections in locally bounded F-spaces with symmetric bases

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVI (1986)

A.

(Poznan)

Projections in locally bounded F-spaces with symmetric bases Introduction. Let X be an F-space, i.e., complete metrizable topological linear space. A closed linear subspace Y of X is said to be complemented (in X) if there is a continuous projection from X onto Y, or equivalently, if there exists a closed linear subspace Z of X such that X is the direct sum of Y and Z, i.e., X = L©Z. X is locally bounded if it has a bounded neighbourhood of zero.

In this note we first show that the theorem of Casazza, Kottman and Lin [3] on operators in Banach spaces with symmetric bases holds in locally bounded spaces. Then we consider separable locally bounded Orlicz sequence

, ,, r M (t) spaces lM with hm- = oo

*0

In this case we prove that every infinite-dimensional complemented subspace X of lM contains a complemented subspace Y which is isomorphic to L

1. Projections in locally bounded spaces. Let X be an F-space with a basis (e„). Recall that a (basic) sequence (x„) in X is said to be a block basic sequence with respect to (en) if, for every n,

Pn

Xn ~ X] ’

i = P n - 1

where 0 = p0 < Pi < ... are integers and (a,) is a sequence of scalars. A basic sequence (x„) is complemented in X if its closed linear span [ x j is complemented.

Two basic sequences (x„) and (>•„) of X are called equivalent provided the

00 *

series £ anx„ converges if and only if the series £ a„y„ converges.

n= 1 n— 1

A bounded set (xx)aeA of points in {X, | * |) is called an г-net if \xx — xp\ > e for all cl , fie A, a # P-

P roposition 1.1. Let (x„) be an e-net in an F-space (X , |-|) with basis (e„).

Then there is a subsequence (x„fc) of (x„) such that:

(i) (>’*) = ( * " 2fc+1- * * 2fc) is a basic sequence;

(ii) (yfc) is equivalent to some block basic sequence {zk) of (en);

00

(iü) Z \гк~Ук\ < 00•

k =

1

Proof. See [7], Proposition II.5.7 or [4], Proposition 3.1.

Recall that if an F-space is locally bounded, then its topology may be given by a p-norm ||-|| for some p g(0, 1] (cf. [7], p. 61). (X , ||-||) is then called a p-Banach space.

A bounded subset of an F-space (X , | |) is precompact if it does not contain an infinite г-net. An operator К: X -> Y between two F-spaces is compact if it maps a neighbourhood of zero in X to a precompact set in Y.

The next Proposition is an easy generalization of a result of Bessaga and Pelczynski [1].

P roposition 1.2. Let (X , ||-||) be a p-Banach space and let (x„) be a complemented basic sequence in X with biorthogonal functionals fx*). If(y „) cz X

00

and Z — ll**|| <oo, then there exists an integer n0 such that (y„)n^„Q is

n = 1

a complemented basic sequence equivalent to (*„)„> „0-

Proof. Let P be a continuous projection from X onto [ x j . We define an operator К : X -* X by

00

K ( x ) = Z xi (Px)(xi - y i), i= 1

and observe that К is compact and (I — K)(x„) = y„ for all n.

The desired result follows from the Riesz theory of compact operators (cf. [6], В. I. § 5).

A basis (e„) of a locally bounded F-space is symmetric if it is equivalent to the basis (хя(п)) for any permutation n of integers.

If X is a locally bounded F-space with symmetric basis (e„), then its topology may be defined by p-norm ||-|| (cf. [5] p. 113) such that

00 (*) ||x|| = sup {||Z ai ai ei\\: W < 1» F C N}> where X = Z ai eL

ieF i

= 1

(**) IkJI = 1 for neN.

00

Thus J : X -» c0 defined by J ( Z

= (ad is continuous and

||Jx||C0 < ||x|| for all x e X .

L emma 1.3. Let X be a locally bounded F-space with symmetric basis (e„)

and let Q be a continuous operator on X. I f JQ(e„) is a non-precompact set in

c0, then Q{X) contains a subspace Y which is isomorphic to X and

complemented in X.

Proof. Since X is locally bounded we may assume that the topology on X is given by p-norm ||-|| satisfying (*) and (**). The assumption of the lemma implies that there exists an infinite set P a N such that |Q(e;): i eP]

and {JQ(ei): i eP) are e-nets in X and c0, respectively.

Applying Proposition 1.1 we can construct two basic sequences (yfc) and (zk) with the following properties:

(a) yk =

i2k+1

(b) zk = X a,c,- is a block basic sequence;

i e A k

00

(c) X

\\zn-yn\\

n= 1

Since \\Jyk\\CQ> £ and ||zk- y k|| ^ ||J(zk- y fe)||Co, it follows from (c) that (d) \\Jzk\\ = max {|af-|: i e A k] ^ rj, where k e N and r\ > 0.

Since for every strictly increasing sequence of integers {nk) the basic sequence (e„k) is equivalent to (en) (cf. [5], Proposition З.а.З), by (c) it follows that (z„) is equivalent to ekn. Moreover, as in [2] we may define a continuous projection R : X -* [z„] by

00 00

R { x ) = X (biJain) zn if ^ = X bnen,

n= 1 n= 1

where |a,J = max {|af|: i e A n\, Vn.

Since sup||z*|| <oo, it follows from Proposition 1.2 that, for some Ио^ЛТ, {yn)„zn0 is a complemented basic sequence equivalent to (z„)n>„0. Thus [>vL>n0 is isomorphic to X and complemented in X.

The next theorem is a generalization of a result of Casazza, Kottman and Lin [3].

Th e o r e m

1.4. Let X be a locally bounded F -space with symmetric basis (e„). Then for every continuous operator Q on X either Q( X) or (I — Q) ( X) contains a subspace Y which is isomorphic to X and complemented in X.

Proof. Since either {JQ(en))neN or {(/ — JQ)(en)}neN is non-precompact in c0 our result follows from the preceding lemma.

2. Projections in Oriicz sequence spaces. An Orlicz function M is a continuous non-decreasing map from R+ to R+ such that M(0) = 0 and lim M( t ) = oo. The Orlicz sequence space lM is the vector space of all scalar t —o

GO

sequences (x„) such that X M{\ex„\) < oc for some e > 0. We define n- 1

00

Вм( е ) = { х : X M ( l * J K s }

n= 1

and then {rBM(e): r > 0, s > 0} is a base of neighbourhoods of 0 for an F- space topology on lM. It is well known that in every separable Orlicz sequence space the unit vectors (en) form a symmetric basis.

Le m m a

2.1. Let M be an Orlicz function satisfying lim M(t)/t > 0, and let t ~*o

(e„) be the unit vectors in lM. An operator A: lM -> f is compact if and only if the set {A(e„): n e N } is precompact.

Proof. Suppose the set \A(e„): neN) is precompact. Since lim M(t)/t > 0, we may assume M( t) ^ t, t ^ 0. Thus

(-o

OO

U = < x = (x„)eU. I M ( W ) < 1 } < = K , n= 1

where

V = {x = (xn) e l M: f W ^ l | . n= 1

Hence F is a neighbourhood of zero in lM. Moreover, A( V) is contained in the closed absolutely convex hull of [A(e^: ne N) .

Let (AT, I * I) be an F-space with a separating dual, and let t 1 be the Mackey topology of X, i.e., the finest locally convex topology weaker than the original topology (cf. [8]). We denote the completion of (X, ij) by X.

Le m m a

2.2. Let X be an F-space with separating dual X* and let Y be an infinite-dimensional subspace of X. I f the operator id: Y X is compact, then Y does not contain an inifinite-dimensional complemented subspace of X.

Proof. If Z a Y is a complemented subspace of X, then X = Z @ W for some subspace W of X. From the assumption it follows that the operator idz : Z q Z is compact. Hence there is a neighbourhood of zero U such that idZ{U) is precompact. Since a closed convex hull of idz{U) is a neighbourhood of zero in Z (cf. [8], Proposition 3), Z is finite-dimensional.

Th e o r e m

2.3. Let lM be a separable locally bounded Orlicz sequence space and let X be an infinite-dimensional complemented subspace of lM. If\imM(t)/t

t-o

= oo, then X contains a subspace Y which is isomorphic to lM and complemented in lM.

Proof. Let P : I m —^I m be a continuous projection on X and let e* be the biorthogonal functionals to the unit vectors e„ in lM. Let us observe that

= f (cf. [4] Theorem 3.3). By Lemmas 2.1 and 2.2, the set A = {P{ei)}ieN is non-precompact in lt . Now, we show that A is non-precompact in c0.

Suppose that our statement is false. Then there exist e > 0 and a sequence vk

= P (ei2k+ l ~ ei2l) SUCh that O O

(0 IKIIq = X \e* M ^ £’

i⁼

1

(ii) IM c0 = sup к? (i?fc)lr ^ O . I

Since the sequence (vk) is bounded in /M, we have 00

(iii) Y j M(|ef(yk)|) < Ô, where <5 > 0 and keN.

i = 1

By (i) and (iii), for each k e N there exists n(k) such that M (Knfa>l) < ôk

Since lim M(t)/t — oo we get inf|e*(k)(ufc)| > 0, contradiction with (ii). Hence

t

-0 к

A is non-precompact in c0 and our result follows from Lemma 1.3.

Remark. Theorem 2.3 does not hold in all Orlicz spaces (cf. [4], p. 276, or [5], Theorem 4.b.l2).

As an easy consequence of Theorem 2.3 and Pelczynski's decomposition technique [5], we have the following result of Stiles [9].

C

2.4. Every complemented infinite-dimensional subspace of lp (0 < p < 1) is isomorphic to lp.

I would like to express my gratitude to Professor L. Drewnowski for his help while working on this paper.

References

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[4 ] N . J. K a lt o n , Orlicz sequence spaces without local convexity, Math. Proc. Camb. Phil. Soc.

81 (1977), 253-275.

[5 ] J. L in d e n s t r a u s s and L. T z a f r i r i , Classical Banach spaces. Sequence spaces, Springer, 1977.

[6 ] D. P r z e w o r s k a - R o l e w i c z and S. R o le w ic z , Equations in linear spaces, P W N , Warszawa 1968.

[7 ] S. R o le w ic z , M etric linear spaces, P W N , Warszawa 1972.

[8 ] J. H. S h a p ir o , Extension o f linear functionals on F -spaces with bases, Duke Math. J. 37 (1970) 639-645.

[9 ] W. J. S t ile s , Some properties o f lp, 0 < p < 1, Studia Math. 42 (1972), 109-119.

INSTYTUT MATEMATYKI

UNIWERSYTETU im. A. MICKIEWICZA POZNAN. POLAND