ISRAEL JOURNAL OF MATHEMATICS, Vol. 26, No. 2, 1977 COMPACT AND STRICTLY SINGULAR OPERATORS ON ORLICZ SPACES

Transcription

1 ISRAEL JOURNAL OF MATHEMATICS, Vol. 26, No. 2, 1977 COMPACT AND STRICTLY SINGULAR OPERATORS ON ORLICZ SPACES BY N.J. KALTON ABSTRACT If 0 < p < 1 and T: Lp(0, 1)--~ E is a continuous linear operator into a topological vector space, there is an infinite-dimensional subspace X of Lp on which T is an isomorphism; thus there are no compact operators on L w Results of this type are proved for general non-locally convex Orlicz spaces. I. Introduction Let 4': [0, co)--> [0, ~) be an Orlicz function, i.e. an increasing function continu- ous at 0, such that 4'(0)=0 and 4'(x)>0 for some x >0. The Orlicz space L,(0, 1) is the space of all measurable functions (identify functions equal almost everywhere) such that for some e > 0 fo' 4'(e-'[x(t)l)dt < ~" L~(0, 1) is an F-space (complete metric linear space) if we equip it with the topology (the 4'-topology) whose base of neighbourhoods of zero consists of all sets of the form rb,(e) where r >0 and B,(e)= x: 4'(Ix(t)l)dt<=e Ifo } for e >0. If 4' is convex then L,(0,1) is a Banach space. (4' is convex if 4' (-~ (x + y))-< 89 4'(y)), x => 0, y => 0.) An Orlicz function 4' is said to satisfy the Az-condition at ~ if sup 4'(2x) < x -i 4'(x) " If 4' satisfies this condition then L=(0, 1) is dense in L,(0, 1). In general we denote the closure of L=(0, 1) in L~(0, 1) by L~(0, 1)./~, consists of all x such that Received November 28,

2 Vol. 26, 1977 OPERATORS ON ORLICZ SPACES 127 fo lqb(aix(t)l)dt<~ for all A>0. An operator T: X ~ Y between two topological vector spaces is compact if it maps a neighbourhood of zero in X to a relatively compact set. Recently Turpin [6] (see also Pallaschke [5]) has shown that if lira x-' qb(x ) = 0 then L,(0, 1) admits no non-zero compact endomorphisms. In response to a question of Turpin ([6] (and [7] 3.4.9)) we shall show that this assumption and the A2-condition at ~ imply that there is no non-zero compact operator T: L~ (0, 1)----~ X where X is any topological vector space. In fact there is no such non-zero operator which is not an isomorphism on some infinite-dimensional Hilbertian subspace of L,. For the special case th(x)= x p, 0<p < 1, this also improves a result of the author [2] that Lp admits no Hausdorff vector topology for which the unit ball is relatively compact. Before going on to the proofs we give a sketch of the technique to be employed. Let sr be a collection of continuous operators with domain/2,. We form on L, the finest topology, Y, such that each T E s~ is continuous. We then construct on //,, a topology /3 stronger than 3' but weaker than the original 4~-topology; this topology has for its base of neighbourhoods of zero the set of 3,-closed ~b-neighbourhoods. Under certain circumstances, it is shown that/3 can be identified as the topology induced by a convex Orlicz function ~ such that i, C L,. Thus each T E sr factors through the inclusion map i, ~/2,. In w we apply this technique to those operators which map {x:ess.sup!x(t)[ <= 1} into a relatively compact set. In w we apply it to those operators which are strictly singular on every Hilbertian subspace of /2,. 2. Compact operators Consider the following classes of operators on the space Lo of all measurable functions on (0, 1) (c,) Sox(s) = o(s)x(s) where 0 E Lo and II 0 I1o = ess. sup f 0(s)l _-< 1. If 0 = x,,, the characteristic function of A, then we write SA for So. (C2) R,x(s) = x(k(s, t)) where

3 128 N.J. KALTON Israel J. Math. K(s,t)=s+t 0<s<l-t =t s=l-t =s+t-1 1-t<s<l. Then each operator of classes C1 and 6?2 defines a continuous operator on L,. In fact if T is in C1 or C2 (where II x II, = 6 (I x (t)l) dt). II Yx In, =< II x II, x E L, Note also that L=, and hence L,, is invariant for each T E C1 or C~. In fact for our applications in this note, we could take for the class (Cz) the larger class of all operators Rx(s)=x(a(s)) where a:(0,1)---~(0,1) is any measure preserving map. The operators R, can be interpreted as follows. Let X be the space of measurable functions on R of period one. Then L0(0, 1) and X can be naturally identified, and R, corresponds to operator x ~ x' on X where x' (s)= x(s + t). (Of course we are ignoring sets of measure zero in this identification.) LEMMA 2.1. Let 31 be a vector topology on I~r weaker than the ~b-topology, but not indiscrete. Suppose each endomorphism of f,, of classes C~ and C2 is y-continuous. Let/3 be the vector topology on I~, whose base consists of all 3,-closed q~-neighbourhoods of zero. Then/3 is the topology induced by an Orlicz function such that tk <-- ~b and 11. II, is lower-semi-continuous for "g. PROOF, For x E L,, define F(x) = inf{e: x ~ B,(e)} (closure in 3'). The sets {x: F(r-lx) _-< e} for r > 0, e > 0 form a base for/3, and F is lower-semi-continuous for y. Clearly we also have F(x)<= n x I1,, and it is therefore enough to show that F(x)= HI x II, for some Orlicz function ~b. We will show that F satisfies the following conditions: 1) If x,y E/~, and ]xl_-<[yl then F(x)<=F(y). 2) If x,,x ~ C,,, O<=x, <=x and x,(t)-*x(t) a.e. then F(x,)--~F(x). 3) If x, y ~/S, and x(t) y (t) = 0 a.e. then F(x + y) = F(x) + F(y). 4) If x E/~,, and 0 < t < 1, F(R,x) = F(x). First observe that T is of class (C1) or (C2), then T(B,(e))CB,(e). Hence as each T is continuous for y, T(B,(e)) CB,(e), and so F(Tx) <-_ F(x) (x E [,,).

4 Vol. 26, 1977 OPERATORS ON ORLICZ SPACES 129 For (1), note that x = Soy where II011 < 1. Hence F(x)<=F(y). For (2), suppose A > 0, I[A(x - x,)[[, = fo' ~b(a(x(t)- x,(t)))dt and 6(A (x(t)- x.(t))) <- 6(Ax(t)) a.e. fo ~ c~(ax(t))dt <~ since xe/~,. Hence by the Lebesgue convergence theorem [[ A (x - x,)ll, ~ 0 for all A > 0, and hence x. ~ x in the ~b-topology. Thus x. ~ x is the y-topology, and hence as F is y-lower-semi-continuous, F(x)<-_liminf.~F(x.). However by (1), F(x)>= F(x.) for all n E N and hence F(x,)-*F(x). For (3), suppose x, y have disjoint supports A and B. Let u. be a net such that u~---~x+y(y) and Ilu~ll,--~F(x+y). Then as SA is continuous for y, SAu, ~ x (~,) and similarly SBu~ ~ y. Thus F(x) <= lim inf~ II S,,uo II. and F(y) _-< lira inl 1[ SBu~ I[,. However and hence If u~ I1o = II s.u~ II. + It sbuo II. F(x + y) >- F(x) + F(y). Conversely if v~ --~ x(y), w~ --~ y(y) and I[ v~ [t, --~ F(x), II wv lie, ~ F(y), then SAv~ ~ x(y) and SBwv ~ y(y). Hence F(x + y) ~ lira inf II $~v,, + Sswv I1~, = lim.!nf (ll sav~ II, + llsow~ I[,) Hence lim inf (ll v~ 11, + H w~ I[,u) a,l, = F(x) + F(y). F(x + y) = F(x)+ F(y). Finally for (4) we observe that R~_,R, = I for 0 < t < 1, and hence F(R,x)~= F(x) < f(r,x).

5 130 N.J. KALTON Israel J. Math. Now we can appeal to 2.4 of [1], with some slight modifications, to deduce that (,) F(x)= ~b([x(t)l)dt Io' for some left-continuous increasing function. For completeness we sketch the details here. Define O(t)=F(txto.~); for fixed r, we may apply the Radon-Nikodym theorem to the absolutely continuous measure A,(A)= F(rXA), to deduce the existence of measurable function 7r~ such that F(,XA)= fa rr,(s)ds A C(0,1). Then since F(R,rxA)= F(rXA) we have F(rXA) = f f 1 ) = m (A)tp(,) x~(g(t,s))rr,(s)dsdt by Fubini's theorem, since f', Xa (K(t,s))dt = re(a). Thus we obtain (*) for simple functions, and we may use (2) to obtain (*) in general. Finally we observe that since 3' is not indiscrete and /3 => % /3 is also not indiscrete. Thus F~ 0 and hence ~ 0, i.e. ~ is an Orlicz function. THEOREM 2.2. Let 4) be an Orlicz function and let T: f~, --~ X be a continuous operator such that T {x:ll x I1~ =< 1} is relatively compact. Then a) If liminfx~= x 14~(x)=0 then T=0. b) If liminfx_= x 14~(x)>0, T may be factorized: f~, -& 1~4, ~ X where c~ is the largest convex function smaller than ck and J is the natural inclusion map. PROOF. Let 3' be the topology on /_~ induced by all continuous operators T:/_],---~ Y where Y is a topological vector space and T(B) is precompact (where B = {x:]] x 1}). If S E Ct or C2, S(B) C B and hence for all such T, TS(B)C T(B) is precompact; therefore S is 3'-continuous. Furthermore B is 3,-precompact. If 3, is not indiscrete we can apply Lemma 2.1 to obtain the Orlicz function $.

6 VOI. 26, 1977 OPERATORS ON ORLICZ SPACES 131 Let {r,} denote the sequence of Rademacher functions (i.e. a sequence of independent random variables with mean zero and taking only the values _+ 1). Since r. E B and B is y-precompact, we may find a net r,,t,)-r,~) where m(a)~n(a) and r~)-r.(.)---~o(y). However [['[1, is y-lower-semicontinuous; hence 1 liminf [[SX,o,,,+~t(r,.,~,- r.(~))[[, >i ~b(s) for s, t >0. However [[ SX,o.,) + ~t(r,.(~,- r.,~,)ll, = 89 to(s) Hence +88 for O<t<=s. 89 O<t<=s i.e. to is convex. Since ~b ~ 49, (a) follows immediately. For (b), note that to is an Orlicz function and that to --< ~ by definition. In fact, although we do not need this, 0 = ~. Clearly T is continuous on/_~, for the q~-topology, for if U is a neighbourhood of 0 in X then T-~(U) is a y-neighbourhood and hence a ~-neighbourhood. Thus the factorization is achieved as required. COROLLARY 2.3. Let 49 be an Orlicz function satisfying liminfx_ O. Then any operator T: L,, --* X which maps bounded sets to relatively compact sets is identically zero. COROLLARY 2.4. An Orlicz space L,(0, 1) has a Schauder basis if and only if 49 is equivalent to a convex Orlicz function and 49 satisfies the A2-condition at oo. PROOF. If 49 is a convex Orlicz function, satisfying the A2-condition at o% then L, (0, 1) has a basis (the Haar system, see [4] p. 101). Conversely if L, has a basis then the sequence {S,} of partial sum operators on L, forms an equicontinuous set of operators and S,x ~ x pointwise. Thus the sets U* = n 7=1 s~l(u) form a base of neighbourhoods of zero, as U runs through the set of all closed neighbourhoods of 0. Each S, has finite-dimensional range and it follows that SII(U) is weakly closed. Hence U* is weakly closed, and /~, has a base of weakly closed neighbourhoods of 0. It follows that to constructed in the theorem is equivalent to 49.

7 132 N.J. KALTON Israel J. Math. REMARK. A further result which follows from Theorem 2.2 is that the weak closure of the set B,(e) is always convex (in /~,). 3. Strictly singular operators If X is a Banach space and Y an F-space, an operator T: X ~ Y is called strictly singular ([3]) if there is no infinite-dimensional subspace E of X such that TIE is an isomorphism. PROPOSITION 3.] (cf. [3]). Suppose X is a Banach space and Y is an F-space. i) T: X--* Y is strictly singular if and only if every infinite-dimensional subspace E of X contains an infinite-dimensional subspace F such that TIF is compact. ii) If T: X--) Y and S: X--~ Y are strictly singular then so is S + T. PROOF. i) Pick a basic sequence (x,) in E; then it is easy to show, using the strict singularity of T, that there is a block basic sequence (u,) with respect to (x.) such that [I u, I[ = 1 and I[ Tu, I[ <= 2-". Let F be the closed linear span of (u,). ii) follows from (i). The proof of the following lemma is omitted: LEMMA 3.2. Let & be an Orlicz function and define qb*(x) = f: &(tx) dt. Then c~ * is equivalent to ck (i.e. L,. = L,); if 4~ is Ck-function (k >- 0), then 4~ * is C k+' on (0, ~). PROPOSITION 3.3. Let & and ~0 be Orlicz functions such that L, C L,. Let y be a vector topology on 1~,, weaker than the 4~-topology such that H" I1, is lower-semicontinuous on ([,,, Y). Then I['11~" is also y-lower-semi-continuous. PROOF. /~, is &-separable. Let 0// be the collection of all y-neighbourhoods of 0. For e > 0, rational B,(e)ns : CI (B,(e)nL,+ U) (closure in 3/). Hence by the Lindel6f property of L,, there is a sequence U~, ~ 0// such that B~(~)ns = (1 (Bo(~)ns u:). n=l

8 Vol. 26, 1977 OPERATORS ON ORLICZ SPACES 133 Let T,, be a metrizable vector topology, To =< T, such that each U~., e E Q+, n E N, is a To-neighbourhood of 0. Then [l" ]1, is T0-1ower-semi-continuous. If x, ~ x (To), then fix [p,.-- fo' ~O*(rx(t)l)dt = fo' s ~(slx(t)l)dsdt 2 -- II sx Ilods s _-< lira inf I[ sx, If* ds _-< lim inf I[ sxo II, as = lim inf ]l x, II," so that I1 tt~ is To-lower-semi-continuous, and hence T-lower-semi-continuous. THEOREM 3.4. Let 49 be an Orlicz function satisfying f~ x249(ax)e-~x2dx <oo A >0. ) Suppose X is an F-space and T: L, ~ X is a non-zero continuous linear operator. Then at least one of the following conditions holds: a) There is a subspace H of IS, isomorphic to an infinite-dimensional Hilbert space such that T IH is an isomorphism. b) liminfx_~ x-'49(x)>0 and Tfactorizes /~, ~ L~, ~ X where ~ is the convex minorant of 4) and J: L, ~ 1~ is the inclusion map. PROOF. Let 3' be the topology on/z, induced by the class ~r of all operators S:/:, ---, Y (where Y is any F-space) such that if R: 12~/Z, is continuous then SR is strictly singular. Let H be any subspace of /:, of infinite dimension isomorphic to a Hilbert space. Suppose T coincides on H with the 49-topology; then there exist maps & :/~, ~ Y~ (1 =< i =< n) in M, and e > 0 such that the set {x ~ H: maxj] S,x ]l =< e} is bounded. Let Y = Y10""" O Y, under the F-norm

9 134 N.J. KALTON Israel J. Math. [I(Y,"" Y-)I[ = max [[y, [[ and consider S : S,@.-.@S.: i,---> Y. By Proposition 3.1, S is strictly singular on H, contradicting our assumptions. Hence 3' is strictly weaker than ~b on H. Next suppose R:IS, ~/:, is continuous; then if S: i, ~ Y is in M, and Q : 12---~ IS,, SRO is strictly singular. Thus SR E M, and R is 3,-continuous. Now suppose 3' is not indiscrete and apply Lemma 2.1. We obtain an Orlicz function q~ whose base of neighbourhoods consists of all 3,-closed 'bneighbourhoods of 0. Then apply Lemma 3.2 three times to obtain a C2-function 0 equivalent to ~b; by Proposition 3.3, I1"11~ is still 3,-lower-semi-continuous. Let (x.) be a sequence of functions in L0, which, considered as random variables, are independent, normally distributed, with mean zero and variance one. Then for A >0 fol ~b(a]x.(t)l)dt = i-~ fo~ $(Ax)e-~2dx so that x, E IS,. For a~ 9 9 <-_ i~ {fo' Cb(Ax)e-]'~dx + f~ x2cb(ax)e-~dx} < ~ 9 a, E R, alx~ a,x. is again normally distributed with mean zero and variance, a] a~.. Hence J] alxl +''" + anx. I1" = II W/~a 2i xl ]1 di~ and hence that there is an embedding W: 12---> IS, with We. = x, (where e, is the unit vector basis of l:). Hence there exists a net Ur = i=1 C i Xi such that u~---> 0(3") but X~-~'~[c712= 1. Thus for s > 0 and e >0, II Sx,o,,llo = 0(s) =< lim inf II SX,o,, + ~u~ I10. However II ~x,o., +,uo Iio -- fo ~ o(i s + ~uo(t)l),~t : O([s+eu(t)l)dt fo

10 Vol. 26, 1977 OPERATORS ON ORLICZ SPACES 135 where u = u(t) is any fixed normally distributed random variable with mean zero and variance 1. If s > 0, since 0 is a C2-function O(I s + ~1)= O(s) + ~o'(s) + 89 ~2o"(s) + ~o(~) where O(0)= 0 and p is continuous. Clearly p satisfies [p(r)l =<A + 0(21r[)_-< A + ~b(16[r D for some constant A. Hence and so O(Is + eu(t)l)dt = O(s)+ fo' O"(s)+ e 2 fo' uz(t)p(eu(t))dt 0"(s) + 2 fo u2(t)p(eu(t))dt>-o. For e =< 1 u2(t)lp(eu(t))j <= uz(t)(a + ~b(16e I u(t)l)) <- u2(t)(a +,/,(161 u(t)[)) and fo' uz(t)(a + cb(161u(t)l))dt=-~ x2(a + cb(16x))e-89 Hence, by the Lebesgue Convergence Theorem, 1 lim uz(t)p(eu(t)) dt = O. e ~0 f0 Thus O"(s)>= 0 and 0 is a convex function. As O(x)<= $(8x)- < ~b(8x), we clearly have O(x) < - 6(8x). Hence if T fails to satisfy (a), then y is not indiscrete, and so 0# 0 and 6# 0. Then liminf x-'tk(x) > 0 x--*~ and T factorizes through L0 and hence through /~,~.

11 136 N.J. KALTON Israel J. Math. COROLLARY 3.5. and such that If qb is an Orlicz function satisfying the A2-condition at liminf x-'oh(x) = O, and X is an F-space containing no subspace isomorphic to 12, then there is no non-zero continuous operator T: L, ~ X. PROOF. The A2-condition at oo implies ~b(x)<--a +Bx p for x--0 and for some p. Hence fo xzrk(ax)e-89 < A>O and the result follows by the theorem. REFERENCES 1. L. Drewnowski and W. Orlicz, A note on modular spaces, Bull. Acad. Polon. Sci. 16 (1968), N.J. Kalton, A note on the spaces Lp, 0 < p = 1, Proc. Arner. Math. Soc., 56 (1976), T. Kato, Perturbation theory [or nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), M. A. Krasnoseloskii and Ya. M. Rutickii, Convex Functions and Orlicz Spaces, Nordhoff, Groningen, D. Pallaschke, The compact endomorphisms of the metric linear spaces L~ Studia Math. 47 (1973), P. Turpin, Opdrateurs lindaires entre espaces d' Orlicz non localement convexes, Studia Math. 46 (1973), P. Turpin, Convexites dans les espaces vectoriels topologiques generaux, Th~se Orsay, 1974 (Diss. Math., to appear). DEPARTMENT OF PURE MATHEMATICS UNIVERSITY COLLEGE OF SWANSEA SINGLETON PARK, SWANSEA, SA2 8PP, U.K.