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Math. Nachr. 119 (1984) 321-326 Approximation by Weakly Compact Operators in L, Dedicated to Professor Albrecht Pietsch on his fiftieth birthday By L. WEIS of Kaiserslautern (Received September 19, 1983) Abstract For every bounded linear operator T in L,[O, I] there is an element of best approximation in the ideal of weakly compact operators. We also give some sufficient conditions for IIT +Sl/ =11Tl1 +i/Sii, where S and T are L1-operators. 1. Introduction It is well known by now, that for every I~ounded operator in lp, 1 sp-=-, there is an element of best approximation in the ideal of compact operators 011 Zp (see [5 Chap. 11, Sec. 71, [6, 10, 11 for different methods of proof) and the same is true for reasonable integral operators in LJ0, 11, 1 -=p-=m (see [ls]). In L,[O, 11 the situation is different : FEDER (in [4]) h&s constructed an integral operator with- out a best approximation in the ideal of compact operators. However, I will show below that for all Li-operators there is an element of best approximation in the spare of weakly compact operators. Let (X, &, p) be a standard measure space. V? denotes the ideal of weakly corn- pact operators in Li(X, &, p). For a bounded linear operator T in L,(X, &, p) put Theorem 1. For a bounded linear operator T in L,(X,&, p) thw? is arb L!(,EV with jlT -Soil =inf {\lT-Sli : SCV} = A($'). For L, operators it seems to be more natural to look for best approximations in V rather than in the space of compact operators. For example, many of the proofs in the Z,-case use thep-additivity of the norm and a formula for the distance to the compact operators similar to A. But in L,(X, 31, p) the corresponding structural properties are linked with weakly comyact operators rather than with compact operators. Furthermore, V' equals the ideal of GOHBERG operators in L,(X, &, p), i.e. % ' is the largest ideal of L,-operators with a compact-like spectral theory (see El41 0 26.6, 26.7 and [13]). There are further differences from the l,-case. There 21 Math. Nadir., Bd. 119

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Page 1: Approximation by Weakly Compact Operators in L1

Math. Nachr. 119 (1984) 321-326

Approximation by Weakly Compact Operators in L,

Dedicated to Professor Albrecht Pietsch on his fiftieth birthday

By L. WEIS of Kaiserslautern

(Received September 19, 1983)

Abstract

For every bounded linear operator T in L,[O, I] there is an element of best approximation in the ideal of weakly compact operators. We also give some sufficient conditions for IIT +Sl/ =11Tl1 +i/Sii, where S and T are L1-operators.

1. Introduction

It is well known by now, that for every I~ounded operator in l p , 1 s p - = - , there is an element of best approximation in the ideal of compact operators 011 Zp (see [5 Chap. 11, Sec. 71, [6, 10, 11 for different methods of proof) and the same is true for reasonable integral operators in LJ0, 11, 1 -=p-=m (see [ls]). In L,[O, 11 the situation is different : FEDER (in [4]) h&s constructed an integral operator with- out a best approximation in the ideal of compact operators. However, I will show below that for all Li-operators there is an element of best approximation in the spare of weakly compact operators.

Let ( X , &, p) be a standard measure space. V? denotes the ideal of weakly corn- pact operators in Li(X, &, p). For a bounded linear operator T in L , ( X , &, p) put

Theorem 1. For a bounded linear operator T i n L,(X,&, p ) thw? is arb L!(,EV with jlT -Soil =inf {\lT-Sli : SCV} = A($').

For L , operators it seems to be more natural to look for best approximations in V rather than in the space of compact operators. For example, many of the proofs in the Z,-case use thep-additivity of the norm and a formula for the distance to the compact operators similar to A. But in L,(X, 31, p) the corresponding structural properties are linked with weakly comyact operators rather than with compact operators. Furthermore, V' equals the ideal of GOHBERG operators in L,(X, &, p) , i.e. %' is the largest ideal of L,-operators with a compact-like spectral theory (see El41 0 26.6, 26.7 and [13]). There are further differences from the l,-case. There 21 Math. Nadir., Bd. 119

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322 Weis, Approximation by Weakly ('ompact Operators in L,

might be a unique element of best approximation in ?% and therefore, the weakly compact operators do not form an M-ideal (see 3.4 and [3]).

The proof of theorem 1 (see 3.1) use8 a formula for A(T) depending on a re- presentation theorem for L,-operators due to N. KALTON (see Rec. 2). This niethnd also gives

Theorem 2. If T and S are bounded linear operators in Ll(X. 32, p) szrch thnt Ti AiSi = O and T'l =ilTll - 1 then

(2) (TFS and JTI refer to the ordering generated by the cone of positive ol)eratora, see e.g. [9]; for some concrete situations with TAS=O, see 3.3.)

Equation (2) has some relevance to approximation theory cf. [15], See. 3. [ 151 also contains a proof for the special case T = Id and P a finite-dimensional projection. The case T=ld , S compact was treated in [2].

jlT + 811 = l j Ti1 + iisii .

2. A formula for d(T)

T n this section we consider measure spaces ( X , 32. p), ( Y , 53, v ) where X and Y are compact metric spaces and & and 1 are the cr-algebras of their BOREL sets.

For a bounded linear operator T : Ll(X,&, p)-+Li( Y , a, v) there are (by [8], [ l l ] 5 5.4) stochaqtic kernels (py)rEY of memure8 on (X, 32) and (vJZEx of measures on ( Y , 9) such that

(3) Tf(Y) =I&) dp&) y-a.e. T'd4 =s g(Y) ( M y ) p-a.8.

for all ] 'cI; , (X, &, p) and gEL,( Y , 83, Y) respectively. LEBESGC~S decomposition !,rovides a useful decomposition T = Ti+ T' of T. We write

py =A +&, pl p-absolutely continuous, pi p-singular

and define

By the RADON-NIKODYM theorem Ti is just an integral operator r+ith kernel

k(y, x) =dpr (x). T* is called an operator with singular representation. Of Course,

there is a similar decomposition for 27' corresponding to

m y ) =/ ' f@) 4&), T6f(Y) = S W d P 3 4 *

dP

ys=vS+vs, i S vk y-absolutely continuous, vl v-singular (4) (for a more detailed discuwion of these decompositions, see [I 71).

In the following statements read 'sup' as 'essential sup'.

Lemma. With the not&ims ( 3 ) and (4) we have

l.5) In; =sup !IvA 2

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Weis, Approximation by Weakly Compact Operators in L , 323

I 0 otherwise.

I n particular: If T has a singular representation, then A(T) =l[Tll. For the proof we need the following result from [17] 2.4 and 5.3.

Proposition. Let S : Lm( Y , Y ) -L,(X, p) have a representation ;Ix 2 0 such that

=$ g(y) d % ( Y ) p-a.e*

Then, for all E > O , there is a closed K C X with p (Kc) z E wuch that

(7) (8) For all K , c K with d(Kn) -0 1) there arc B, c Y with v(B,) - 0

si, B8 : C( Y ) - C ( K )

and

(1 +&) s B ( x B n ) Z x K n * ss(l) Proof of t h e lemma. The first statement is well kilonil anti f o l l ~ ~ u s from

IlTII=IIyII=II ITI’1Ilm=suPJ1d lvzl * X

Put 7 1 : =inf sup { ~ l v ~ ~ ~ + J jkn(y, x)l d v ( y ) and n x E 9

kqJ(x)=.ffd I4I+.f Ikn(yt %)I f ( y ) dv(y), fEL-(Yi v) * Theti q=inf llSnll. In order to show that d(T) srj we choose B,c Y such that

n

y ( B m ) -0 alld Ilxlt,TII -d (F ) - For a fixed ‘n we have

iIxBmFII =SUP { I Y ~ I ( B m ) + J XEX Bm

x)I d v ( y ) I

SSUI’ { lY; I (y)+s Ilc,(y, 41 d v ( y ) + 4 4 ? 2 ) . X €X

For m-00 we obtain d ( T ) sIIAYJ.

to 8,. Hence there is a KcX satisfying (7), (8) and It remains to show that d ( T ) ~ q . Fix E=-O and nCN and apply the 1)rofiwition

(9) ~IxdvzIl >llsnll --E *

Choose an xoCK such that &l(x)>IIx9SJ - 8 in a neighborhood nf x,,. By (7). (8)

1) d(K,) denotes the diameter of the set K , wit,h reapect t.0 t,he metric of Jy. ? I *

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324 Weis, Approximation by Weakly Compact Operators in L,

1

n

%I ( E ~ = % ( x E ~ ) ( x ) = - ~ C ~ ( X O ) -s=li401i --F

there is an E+,c J' with v(E,) z - such that

($0)

in a neighborhood of xo (choose the K , ai3 neighborhoods of xo). Since

1

n there is an F , ~ { l k + , ( * ~ xo)l

(12)

in a neighborhood of xo. Put B, = E n u F,. Together (9) to (12) imply

with v(F,) s 2ilTIJ such that

J lk.n(Y9 2) M Y ) SJ lIc,(Y, x0)l d4Y) - 8 Fa

3. Proof of the Theorems and Remarks

3.1. Proof of Theorem 1. It is well known that S : L, -L , is weakly corn- 1)act if aridonlgifS(ULi)isequi-integrable,i.e. ifandonlyifd(S)=O(see[9]§ 6.11). Therefore, we have for all S€?? that

The proof is complete if we find an fro€% with JjT--S0jl=d(T). To this end w e again use the representation (3), (4) : For all

we choose t,€R+ in such a way that for B,=(y : Jk(x, y)) >t,> we have (13) Ib'ilI + Ik(y7 dv('!/)=d(T) .

lyP-S\\zLl ( T - S ) = d ( T ) .

xEA: =(xEX: \]YJ\=-Ll(T)]

Bz { t(g, 5) for ye&, ~ E A Define h.(y, x) = otherwise. The operator 8,, defined by

satisfies IIT -So\i = A( 2') by (13) and the lemma. It remains to show that Otherwise. there is some C>O such that for all nEX

f io f (Y) =s k(% 4 f ( 9 ) f a x 7 P ) = 0.

sup J' h&, x) dv(y) zc % € A

(14)

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Weis, Approximation by Weakly Compact Operators in Li 325

If hs(., 2) + O then Bxc{Jk(-, x)l =-%I and we obtain with (13)

II4I+J Ikac,(Y, 41 W Y ) zII4ll+ J’ Ik(?/, 41 W?/) +J’ I U Y , 41 W Y )

=om+.!- lh,(Y? 41 d4Y) * BX

Kut together with (14) this leads to the contradiction d ( T ) & d ( T ) + C . 3.2. Proof of Theorem 2. Let T and x‘ be represented (in the sense of (3))

hv stochastic kernels ( v ~ ) , ~ ~ and (A,JZEX, respectively. Then (T 4- 8)’ is represented by (&+v, ) ,~~ . The assumption \TI A(XI = O implies (cf. [17 2.3) that

141 A lvxJ = 0 p-a.e. and T‘1 =((TI1 1 means that

l l ~ x I l = l ~ z l (1)=lTl’1(~)=llTll 1(X)=IITII .

II 4c + %ll = 11~A1 + l l % l l = II I + I I TI1 .

Now the claim follows from the lemma and

3.3. Remarks. The assumption T AS = 0 of Theorem 2 i R fulfilled for example if i) S is an integral operator (e.g. X compact or weakly cornpact) and T has a

singular representation (e.g. T a RIESZ homomorphism or a convolution by a singular measure),

ii) X has a diffuse and T an atomic representation (see [17]).

3.4. Remark. Tf T hag a singular representation and TI ==IITII 1 (e.g. T =id), then the zero operator is the only element of be& approximation in V . Indeed, if N is weakly compact it follows from 3.34 and the lemma that

ilT+8/1= ilTIl+ IlXlI = + IISII . But then [el, theorem 3, implies that V is not an 1M-ideal in the space of bounded linear operators (this was already observed in [3]).

3.6. Remark. In [ I 21 it is shown that for an operator 2’ : L,(X, p) -L,(X, p) there is a compact operator So in L,(X, p ) such that

i(T -6,il =in€ (\IT -811 : S compact}

nlzd

11(T-S0)211=inf fllT’--SII : S compact) . There is no analogue for this in our situation. Let il. be a measure on the cirde T such that p is singular with respect to LEBEsGuE-meamre but p * p is absolutely continuous with respect to LEBESGUE measure (for the existence of such measures see [7] Chap. VIII, Theorem 117). Then the convolution operator ’r f=f*i l . in L,(T, p) has the singular representation ,uz(A) =il.(x-i * A ) and by 3.4 the zero operator is the only element of best approximation to Tin T. But I/T”l i A(T2) = 0.

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326 Weis, Approximation by WPakly Compact Operators in Li

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