OPERATORS INTO Lp WHICH FACTOR THROUGH /,

WILLIAM B. JOHNSON

ABSTRACT

A bounded linear operator T into Lp[0,1] (2 < p < oo) factors through lp if and only if T iscompact when considered as an operator into L2 [0,1 ].

I. Introduction

In [3] E. Odell and the author proved that for 2 < p < oo a subspace of

Lp(= Lp[0, 1])

either contains a subspace isomorphic to l2 or embeds isomorphically into lp. Inother words, if T is an isomorphism from a Banach space X into Lp, and /2 is notisomorphic to a subspace of X, then T factors through some subspace of lp, i.e., thereare a subspace Y of /p and (bounded, linear) operators A : X -*• Y, B : Y -*• Lp sothat BA = T. The proof in [3], however, does not yield that T factors through lp

itself.In the present paper we classify the operators into Lp (2 < p < oo) which factor

through lp. It is well known (cf., e.g., the appendix to [9]) that every operator fromlp into L2 is compact; so a necessary condition for T : X -* Lp to factor through lp

is that T should be compact as an operator into L2. In §11 we prove that this conditionon T is also sufficient for T to factor through lp. Now it follows from [4] (cf. [3])that if T is an isomorphism from X into Lp, then T is compact as an operator into L2

if and only if X has no subspace isomorphic to /2; so the theorem stated in the abstractimproves the main result of [3] mentioned in the preceding paragraph.

We use standard Banach space theory terminology as may be found, for example,in [6].

II. Factoring operators into Lp, 2 < p < oo

Throughout this section, T is an operator from some fixed Banach space X intoLp (2 < p < oo) and T is compact when considered as an operator into L2. Wewant to prove that T factors through lp. Actually, we factor T through a space ofthe form Y = (J^(Hn, |-|n)),p, where each space (Hn, \-\n) is finite-dimensional. Weshall observe that the spaces (Hn, |-|,,) are uniformly isomorphic to uniformly comple-mented subspaces of Lp, and hence Y is isomorphic to a complemented subspace oflp. (Of course, this implies that Y is isomorphic to lp by a result of Pelczynski [8],but we do not need this fact, since it is clear that if T factors through a complementedsubspace of lp, then T factors through lp.)

The spaces (ff,,) are chosen to be a blocking of the Haar basis for Lp. That is,Hn = span (hd^^-1, where (/*,) is the Haar basis for Lp and 1 = k(l) < k(2) < ...is a suitably chosen sequence of positive integers. The operators A : X -*• Y and

Received 21 April, 1976.

Supported in part by NSF-MPS 72-04634-A03.

[J. LONDON MATH. SOC. (2), 14 (1976), 333-339]

334 WILLIAM B. JOHNSON

B : Y -> Lp are defined in the natural way: for xeX with T* = %yn (yneHn), wedefine ,4* = {yj»ml. For j/neifn with (ytfml e 7, we define JB(yJ = 2>neLp.Obviously we have BA = T, but of course we have to show that A and B are boundedif the (Hn, |-IJ sequence is appropriately defined.

The reader who is familiar with the proof in [3] will observe that the approach in[3] for the case when T is an isomorphism was similar. In [3], an appropriate blocking(Hn) was defined, |-|n was the usual Lp norm restricted to Hn, and A was defined as inthe preceding paragraph. Now Hn is norm 2 complemented in Lp, since the Haarfunctions form a monotone basis for Lp; so Y = (Z,Hn)lp is isometric to a norm 2complemented subspace of lp. The operator A is continuous no matter how theblocking (Hn) is defined, and the hypothesis that T is compact as an operator intoL2 allowed us to define the blocking (Hn) of the Haar system to make A an isomor-phism. This means that the operator B defined in the preceding paragraph is con-tinuous when considered as an operator from AX into Lp. However, B cannot bebounded as an operator from Y to Lp when each |-|n is the usual Lp norm.

In the present paper, |-|n is defined in such a way that we must check all threethings: that Y is complemented in lp, that A is bounded, and that B is bounded.

It is convenient to define |-|B on all of Lp. For appropriate values of Mn, 1 = Mx <M2 < M3 < ..., |-|n is defined by

|/ln = max (MJ|/ | | 2 , | | / | | P) ,where

/i \i

B/lla s Ml/(OI2«M.

have their usual meaning. It is evident that each |-|n is equivalent to ||-||p on Lp, butas Mn \ oo the constant of equivalence tends to infinity.

We break the proof that T factors through Y if (Hn) and (MJ are defined appro-priately into three steps.

Step One. There is a constant K = K(p) such that {Hn, |«|J is K-isomorphic to aK-complemented subspace ofLp.

Of course, this means that Y is isomorphic to a complemented subspace of lp nomatter how Mn is defined.

Step one is easy, given a result of Rosenthal [10]. Rosenthal proved that there is aconstant A = A(p) so that for any sequence w = (wlt w2, ...) of positive numbers thespace XPt w is A-isomorphic to a A-complemented subspace of Lp. Here XPt w is thecompletion of R°° (or C00) under the norm \\-\\w defined by

It is easy to see that (Hn, |-|n) is isometric to a norm 2 complemented subspace ofXPi w for some w. Indeed, since each element of Hn is a step function and dimHn < oo,there is a sequence (even finite) of disjoint intervals (A() so that Hn c span (%Al). Let

" (= HxJI a/IIX*U

OPERATORS INTO Lp WHICH FACTOR THROUGH lp 335

and set/, = (meas A{) llPxAi (so that ||/,||p = 1). Then, for any choice (a,) of scalars,

wt2)\

i.e., spanx^, is, in the |-|n norm, isometric to XPtW when w = (Mnwx, Mnw2, . . .).Thus, by Rosenthal's theorem, we can complete the proof of step one by observingthat (Hn, \-\n) is norm 2 complemented in (Lp, |-|n) and hence in span;^ . But theorthogonal projection P onto Hn satisfies | |P| |2 = 1 and (since the Haar functionsare a monotone, orthogonal basis for Lp) ||P||P < 2; hence \P\n < 2 by the definition

ofl-L.

Step Two. B has norm < 13 provided that, given Hlt H2, ..., Hn, Mn+2 is chosensufficiently large.

Suppose that the blocking (Hn) of the Haar functions and numbers (Mn) are given.We want to compute that for yn eHn, ||5>B||P < 13Q2 \yn\Z)l/p, as long as each Mn+2

is big relative to the modulus of uniform integrability of Ht+ ... +Hn.Let M = {n : \yn\n ^ 2nI>;J|p}. Certainly

III ym li Z ynnfM

+ I bn\

so we need check only that

and

p n e M

Vln

I y, +(Z.\yn\p)1/p>

2 n - l

Forn$M we have that Mn||yn||2 < \yn\n ^ 2n\\yn\\p) so that < 2" Mn

Now if 2" Mn * is very small, this means that yn is essentially supported on a set ofvery small measure; hence, if y is a fairly flat function in Lp, then

Thus, if Mn+2 is chosen big relative to the modulus of uniform integrability ofHx+ ... +Hn,then

y2, \2n t M

/p

This kind of reasoning is used (among other places) in [3] and [2]; we reproduce anargument in the proof of Theorem III.2 of [2] for the convenience of the reader whowishes to check the technical details.

For notational convenience, let K = {1} u {« : n $ M and n is odd}. We want toshow that

neKA Z 11*11? >

336 WILLIAM B. JOHNSON

J

as long as Mn+2 grows fast enough. Set st = \. Given H = Hl+ ... +Hn, chooseen+! > 0 with 2en+1 < en so that iffeH and E £ [0, 1] with meas £ < e^T1 } , then

i/p

i/rl

\E /

Now assume that 2"+2 M~+2 < 2"( n + 2 ) pen+i; i.e., that Mn+2 is small with respect tothe modulus of uniform integrability of Hx 4- ... +Hn.

Set £x = [0, 1]; for neK with « > 1 set En = {t: |^,(0l > e i - l~ p ) Ibnllp). and letOO

Fn = En ~ (J £,-. (Here and in the next two paragraphs unions and sums are«=n+l

taken over indices in K.) Observe that, by the definition of En,

meas Fn ^ meas£n < e,^-i~1),

so that meas ([0, 1] ~ (J Fn) = 0 because £e p / ( p ~ 1 ) < oo. Thus

i/p

| p

•fn

n - 2

+\y*\'+i=n + 2

I /P

S y,

1/P1

= 3[a+b + c].

Observe that, by Schwartz's inequality, we have for n > 1 that

i / p l/2p

•>-%

lnMn-'\\yn\\pYIPt~n^

Letting Dt = [j Fn, we thus have that

" \\yn\\p.

1-2un = 1

C =

00

1 = 3

1/P

•( /

T yiXD,i = 3

I / P

Of course we have ft ̂ (X ll);n||p)1/p> so it remains to estimate a. Since

measF,, ^ e^r1*, and ^ 4 - ... +j/B_2eH1+ ... 4-Hn_2,

OPERATORS INTO Lp WHICH FACTOR THROUGH lp 337

we have from the choice of en and the monotonicity of the Haar basis for Lp that

n-2v - l

n = 3

Therefore

max

Z2 n - l

A similar argument shows that

hence

3-2(1 ||^

13(Zas desired.

Recalling that the blocking Hn = spansequence 1 = A:(l) < k(2) < ..., we state

^ ^ " 1 is denned by the increasing

Step Three. A has norm ^ K\\T\\ (where K = Kp is a constant which dependsonly on p) provided that, given Mn(n > 1), k(n) is sufficiently big relative to Mn.

Let || S || 2 be the norm of the operator S when considered as an operator into L2.Let Rn be the orthogonal projection from L2 onto span (/*j)?Ln in L2. Our hypothesisthat T is compact as an operator into L2 implies that ||.Rn T| |2-> 0 as n -* oo.Suppose now that \\Rk(n)T\\2 < 2""M,,"1 | T | | for n = 2, 3, .... For xeX with

Tx= yn (yneHn\

we need to show that

Let M = {n : \yn\n = | | ^ n | | p } . Since the Haar system forms an unconditional basisfor Lp [7], there is a constant 0 < X = X(p) so that

(cf., e.g., the argument on p. 209 of [9]); thus

i/p / \ i/p/ \ 1 / p /

\neAf / \n eM

Observing that 1 e M (since M± = 1), we have that

cl lyjj?i /p

Z Mnk=n 2 n = 2

338 WILLIAM B. JOHNSON

Thus

( 00 1/p

as desired.Of course, to complete the proof that T factors through lp, we only have to make

the obvious observation that the sufficient conditions in steps two and three for theboundedness of B and A are not mutually exclusive.

Remark 1. The ideas used in steps two and three are contained implicitly in [3]and [2]. The main new idea here involves the use of the weighted L2 norm in thedefinition of |-I,,.

Remark 2. The interesting thing to me about the argument presented here is thatRosenthaPs not-yet-classical space Xp plays a fundamental role in the proof of atheorem whose statement mentions only classical spaces.

Remark 3. It is very easy to prove (cf. [1]) that every compact operator into Lp

factors through lp. Say that an operator into Lp satisfies (*) provided that eachE £ [0,1] with meas E > 0 contains a subset F such that meas F > 0 and the operatorXF T (defined by (xF T)x = xF.(Tx)) is compact. A standard measure-theoreticexhaustion argument shows that if T satisfies (*), then T factors through lp. Ouroriginal approach to the problem discussed in this paper was to show that if T : X -+LP

is compact as an operator into L2, then T satisfies (*). This is not true, however;in fact the basis-to-basis map from the unit vector basis of lt to the normalized Haarbasis for Lp fails (*). It is even possible to construct an isomorphism from \p intoLp which fails (*). Perhaps the collection of operators into Lp which satisfy (*) is aninteresting class which is worthy of study.

Remark 4. Of course, the dual version to the result proved here is that anoperator T with domain Lq (1 < q < 2) factors through 1q if and only if T is compactwhen considered as an operator fromL2. This is false for q = 1. The proofs in [5]yield that an operator T from Lt factors through \y if and only if T is differentiate ifand only if T satisfies the dual condition to (*)—i.e., every set E £ [0, 1] of positivemeasure contains a subset F of positive measure so that the restriction of T to L^F)is compact. On the other hand, J. Uhl has observed that an operator T from Lt is aDunford-Pettis operator if and only if T is compact when considered as an operatorfrom L2, but it is known that there are Dunford-Pettis operators on Lx which arenot differentiate.

Remark 5. We conjecture that an operator T from Lp (2 < p < oo) factors throughlp if and only if T is of type p Banach-Saks; i.e., there exists a constant A suchthat every weakly null normalized sequence in L_ has a subsequence (xn) for which

Txt f o r / i =

This conjecture was verified in [2] in the case when T has closed range.

Remark 6. There is a non-separable analogue to the theorem stated in the abstract.Suppose that 2 < p < oo and T : X -> Y - (£a 6 A Lp(/ta))/p(/1) is an operator, whereeach jia is a probability measure. (It is well known that every Lp(/i) space is isometric

OPERATORS INTO Lp WHICH FACTOR THROUGH lp 339

to such a 7.) For each a e A, let Ta be the composition of T with the projection from

y onto Lp(/ta). Then T factors through

if and only if each Ta is compact when considered as an operator into L2(na).

References

1. W. B. Johnson, " Factoring compact operators ", Israel J. Math., 9 (1971), 337-345.2. , " Quotients of Lp which are quotients of/p ", Compositio Math., to appear.3. and E. W. Odell, " Subspaces of Lp which embed into /p ", Compositio Math., 28 (1974),

37^9.4. M. I. Kadec and A. Pelczynski, " Bases, lacunary sequences, and complemented subspaces in the

spaces Lp ", Stadia Math., 21 (1962), 161-176.5. D. R. Lewis and C. Stegall, " Banach spaces whose duals are isomorphic to U (T) ", / . Functional

Analysis, 12(1973), 177-187.6. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Lecture notes in Mathematics 338.

(Springer-Verlag, 1973).7. R. E. A. C. Paley, " A remarkable series of orthogonal functions ", Proc. London Math. Soc,

34 (1932), 241-264.8. A. Pelczynski, " Projections in certain Banach spaces ", Studia Math., 19 (1960), 209-228.9. H. P. Rosenthal, " On quasi-complemented subspaces of Banach spaces, with an appendix on

compactness of operators from Lp(n) to Lr{v)", J. Functional Analysis, 2 (1969), 176-214.10. H. P. Rosenthal, " On the subspaces of Lp (p > 2) spanned by sequences of independent random

variables", Israeli. Math., 8 (1970), 273-303.

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