1Й1 112/J < +°°,(

ANNALES SOCIETATIS MATHEMAT1CAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I V (1970) R O C ZN IK I POLSKIEGO TO W A R ZYSTW A MATEMATYCZNEGO

Séria I : PRACE MATEMATYCZNE X I V (1970)

A. P

(Warszawa)

O n Sikorski’ s trace formula

The purpose of this note is to present a proof of the following result (the terminology and notation are explained below)

T

. Let A, B , G be 2-absolutely summing operators on a Banach space X . Then

( 1 ) Ьгво{А) = tvA B (C) = Ьхал(В ).

Formula (1) was conjectured by Sikorski [7] and was proved by Grothendieck [1]. Sikorski originally assumed that the operators A , B, G are integral. Since every integral operator is 2-absolutely summing (cf.

[в]? [3], [6], [2]), our assumption is less restrictive.

We recall that a linear operator N : X -> Y (X , Y — normed linear spaces) is nuclear if it has a nuclear representation

X = ^ xn ® yn, П

i.e. there are sequences {x*) of bounded linear functionals in X* — the dual of the space X , and (yn) of elements of У such that

1Й1 112/J < + °°,

( 2 )

N(x) = J ? Яп(х )Уп f° r x in X . П

We put

q(N) = g.l.b. £ \K\\ \\Vn\\i n

where the g.l.b. is taken over all nuclear representations of the operator N.

If N = x* ®уn: X -> T is a nuclear operator and A : T X П

is a bounded linear operator, then we write tr^v (A) = x l ( A { y n)).

П

It is well known [8] that the quantity tr ^ J .) depends only on N;

120 A. P o lczy n ski

it does not depend on a particular choice of a nuclear representation of N.

We have

(3) |trlV( i ) | < S(J)||i[|.

Next we recall that a linear operator A : X -» Y { X, Y — normed linear spaces) is said to be 2-absolutely summing if for some (equivalently, for all) isometrically isomorphic embedding JA from X into a space C(Q a ) there exist a probability measure mA on QA and a bounded linear operator SA : Li(mA, QA) -> Y such that

-d $

^

7

where : C(QA) -> L 2(mA , QA) is the formal identity map which assignes to a continuous function / in C(QA) its mA-equivalence class in L 2(mA, Qa ) (cf. [5], [2], [3], [6]. In [2] and [3] instead of “ 2-absolutely summing” the term “ 2-integral” is used).

By G(Q) we denote the space of scalar-valued ( = real-valued or complex-valued) continuous functions on a compact Hausdorff space Q.

If m is a probability measure on Q ( = a regular non-negative Borel measure of total mass 1), then L 2(m, Q) denotes the Hilbert space of m-equivalence classes of square-integrable scalar-valued functions on Q).

Observe that the set of all 2-absolutely summing operators from X into Y forms a linear manifold in the space of bounded linear operators from X into Y (cf. [6], [5]). Furthermore note that the assumption of our theorem implies that the numbers Ь твс {А),Ь та в (С),Ь тса (В) are well defined, because the composition of 2-absolutely summing operators is a nuclear operator. More precisely we have:

L

1. Let JB = S в I тв Jв • X — > Y and G — So I me Ac : Z — > X be 2-absolutely summing operators, where X , Y, Z are normed linear spaces,.

J b : X - > G { Q

) and J o: Z - + C { Q c ) are isometrically isomorphic embed­

dings, I mB: G (Q

) - > L 2(mB, QB) and I mc: C(QG) L J mc , Qc) are formal identity maps, SB- L 2(mB, Q

) Y and Sc : L 2(mc, Qc) -> N are bounded linear operators. Then the operator

G(Qc) -> Lz(mB,QB) is nuclear and

(d) < 1 ( 1 твТ в 8 с 1 тс) < л а д .

This lemma is an immediate consequence of [4], p. 56, Satz 2 and the fact that the operator I mB(JBSc): X 2(me , Qc) L 2(mB, Qв) is a Hilbert-Schmidt operator with the Hilbert-Schmidt norm equals to the operator norm of Sc (cf. [4], p. 56, Satz 1).

Our next lemma reduces the proof of the theorem to the case where

X is a G(Q) space. For a given Banach space X, K* denotes the unit ball

SikorsM ’s trace formula 121

of the dual space X* equipped with the weak-star topology and J: X -» C(K*) denotes the eannonical isometrically isomorphic embedding defined by

J{x){x*) = x*(x) for x z X and for x*eK*.

L

2. Let A = SAI mAJ , В = SBI mB J ? C = ®cImcJ be 2-absolute- ly summing operators on a Banach space X . Then

A J 8 A ^-mA ) B = C = J8BI mc are 2-absolutely summing operators on C(K*) and

tr.BC (A) = tvè ô {Â).

P r o o f . Observe that A = I mj J j- , where J j is the identity operator on C(K*), mA = m j and S j — JSA- Hence the operator A is

-absolutely summing. The same argument shows that the operators.

Ê and C are

-absolutely summing.

To prove the second part of Lemma

we observe first that Lemma

implies the nuclearity of the operator

N = SBI mBJSc I mo: C(K*) - + X . Hence it has a nuclear representation

N = ^ m n®xn -, mne(C{K*))* , xnc X {n =

,

, . . . ) . П

Since JN — BC and NJ = BG, we get the nuclear representations.

BC = 2 mn®J{ xn)l BC = 2

П П

where J*: (C(IT*))* X* denotes the adjoint operator of J. Thus, using the obvious identity JA ~ ÂJ, we get

t r j B c ( ^ ) = 2 (J*(тп))(ЛЫ ^{) =} 2 mn(JAiXn))

n n

= 2 mn(A J (Xn)) = tr^

(

).

n

This completes the proof of Lemma

.

In the sequel we shall assume that X = C{Q). In this case if A : G(Q) -> C{Q) is a

-absolutely summing operator, then there is a probability measure on Q, say mA, and a bounded linear operator SA: L 2(mA , Q) -> C(Q) such that A = 8AI mA (we take for JA the identity operator on G(Q)). In fact, we have the following representation lemma:

L

3 . I f A = SAI mA: C(Q) -> C(Q) ia s 2-absolutely summing

122 A. P e lc z y n sk i

operator, then there exists a set of indices stf, an orthonormal system {aa)ae^

in L 2(mA, Q) and a family {aa)ae^ in C(Q) such that

(5) ll^ll = SUP ( ^ \йаШ2)У2’

<6) A ( f ) = f fd a dmA • aa.

аел/Q

P r o o f . Let (aa)aej# be an arbitrary complete orthonormal system in L 2{mA l Q)-

Then

I mA( f ) = / / « « dmA-aa for feC(Q).

aejtfQ Thus

A { f ) = SAI mA (f) = f fd a dmA ’ 8A{aa).

W e put cia = SA (««) for azsé. Then clearly we have (

). Finally we have

\\S a \\ = sup sup I V tadaiq)I = sup sup I У y<xa( d = sup ( V |ae(ff)|*)%.

2\tal* = l aeQ aejtf ' VQ 2\ta\'^ 1 aTS'

This completes the proof.

Our next lemma is a slight modification of [4], p. 56, Satz

.

L

4 . Let m be a probability measure on a compact Hausdorff space Q. Let (bp)peS$ be an orthonormal system in the space L 2(m, Q) and let (Cy)ye<e be a family of functions in C(Q) such that

(7) sup QT

M g ) l 2)*' = s < + ° ° -

q e Q _{y e <jf}

Then

( 8 ) ( 2 2 \ f cy(q)b/3(q)dmj2jy2 < *•

P e @ y e # l Q

P r o o f . Let Z2(^) denote the Hilbert space of scalar-valued functions t = {ty)ye<# such that |[t|l2 = £ \ty\

< -j— oo. It follows easily from (7) that

y e #

the formula

8(1) = £ t yCy for teZa(^).

y e #

defined a linear operator from Z2(^) into C(Q) with the norm \\S\\ = s.

How, if I m: G(Q) -> L 2{m, Q) denotes the formal identity map, then,

Sihorslci’s trace formula

123

by [4], p. 56, Satz

, the composition I mS is a Hilbert-Schmidt operator and its Hilbert-Schmidt norm is < ||$||. Thus, if (ey)ye<Sf denotes the unit- vector basis in 1г(Щ, then using the Parseval identity we get

* = IISII» (y>„S<e,)II*)* = Р Л Г )* 4

У s ' # y e #

V e V P e Æ Q

This completes the proof of the Lemma.

Now we are ready for the main lemma.

L

5 . Let В = 8в 1тв and G — 8 c l mc be 2-absolutely summing operators on G(Q) and let (according to Lemma 3) for fcC(Q)

B (f) = ^ f fbp dmB -bp', 0 ( f ) = Jfcydmc-Cy,

P e @ Q y t V Q

where (bp)p€& and (cY)Ve<g are orthonormal systems in L 2(mB,Q) and in L 2(mc , Q ) respectively, bpeC(Q) for (SeB and cyeC(Q) for y**#, and

sup 2 Ib« («)l

= ll-Sull2; sup 2 lcr

= ll*cll2.

P e Æ Q* ® y e #

Then for every bounded linear operator A : C(Q) -+C(Q) we have

(9) tr BO(A) = E E S iybpdmB J A(bp)cvdm0 .

y e # Pe& Q Q

P r o o f . It follows from Lemma 4 that the double series in the right- hand side of (9) is absolutely convergent because for every q in Q we have

у к^(ь„))(8)|г < m r y ; i m «) i s< над» и р .

ре@ РеЗЗ

Hence for e >

one may choose finite subsets of J* and of so that

(

) 2 E S * pdmB j A(bp)Cydmc—

y e # PtSSQ Q

- E E S cybfsdmBj A(bp)cydmc \< e,

Y*V0 fic# 0 Q Q

(

) sup . 2 |Ь/?(з

<'- sup |Cy((Z)l

<C £2.

Pe£ÿ\&0

124 A. P e lcz yn sk i

Let us set for feC(Q), S b 0 (/)

Next put

f fb(}dmB-bp;

P e

&0

Во Вв^1тВ1 It follows from (11) that

So0 (/) = f fCydmc-Cy.

y^o Q

Go = SoJ.. ₀ mc ‘

(12) ||$ b — $j?0ll < £j \\Bo~ $ c0ll < £-

Next observe that for a fixed bounded linear operator A : G(Q) -> C(Q) the quantity trBa(A) is a bilinear form on the linear manifold of all 2-absolutely summing operators on G(Q). Indeed, if mn®fn and U Pn® 9n

П П

are nuclear representations for operators BC1 and BC2, respectively ( В , С Х, С 2 — 2-absolutely summing operators on C(Q)), then ^

71

+ 2 Pn®9n is a nuclear representation for the operator В(СХ+ С 2). Hence П

tr щс 1+ с 2){А) = ^ m n( A { f n)) + ] ? p n(A{gn)) = tTBOl{A) + Ьт Bc2{ A ) .

A similar argument shows that tr^ + b y c (A) = t r ^ (A) + t r ^ e (A).

Thus, observing that ||# b 0|| < \\S b \ \ and using inequahties (3), (4) and (

), we get,

(13) |tr^o(A) — trj5oCfo(A) I < \^(B- bq ) o {A)\ + |trBo(C'_r

)(A)|

< ( ^ ( ( Б - Б о)С) + д(Б

( 0 - О о)))||А||

< ||A|| \\8o\\ I l ^ - ^

II+||A|| \\SBo\\ \\S o - S c 0\\

^\\A\\(\\8c \\+\\SB\\)e.

(observe that the operators C0 and B0 are finite dimensional and therefore they are

-absolutely summing and nuclear).

Since the sets and &0 are finite, the formula

B0G0(f) = I A J > ydmc f cyb[idmB-bp for feG{Q) У€<&0 Q Q

gives a nuclear representation of the operator B0C0. Hence (14) tr£ocf

(A) = £ £ f A ( b fi)cydmc f tvb^dmB.

y*V QPeâfQ Q Q

Combining (

), (13) and (14) we get

I tre e (A) — S S I A{bp)Cydm0 f CybpdmB\ < e(l+||A||(||tfB||+ ||£ c ll))-

ye''£

Sikorski’s trace formula

125

Thus, letting e tend to zero, we get (9). This completes the proof of the lemma.

P r o o f of t he t h e o r e m . Let A : C{Q)-^C{Q) be a 2-absolutely summing operator satisfying conditions (5) and (

) of Lemma 3. Let В and G be as in Lemma 5. Then, combining (

) and (9), we get

(15) Ьтв с (А) - I I S ^tvbpdmB J f bpaadmA -a^Cydinc

yetf РеШ Q Q « 6 ^ Q

= 'У , У ! У ! f CybpdmB fbpdadmA f ^{aaCydmc .}

PeâS Q Q Q

(by Lemma 3 applied to the “ representations” of A, В and 0, all series appearing in formula (15) are unconditionally convergent). Since a cyclic permutation of the triple (A, B , C ) does not change the right-hand side of (15), we get (1). This completes the proof of the theorem.

References

[ 1 ^] A . G rroth en d ieck , The trace of some operators, Studia Math. 20 (1961), pp. 141-143.

[2] A . P e l c z y n s k i , p-integral operators commuting with group representations and examples of quasi-p-integral operators which are not p-integral, ibidem 33 (1969), pp. 63-70.

[3] A . P e r s s o n and A . P ie t s c h , p-nukleare und p-integrale Abbildungen in Banachraumen, ibidem 33 (1969), pp. 19-62.

[4] A . P ie t s c h , Nukleare lokalkonvexe Baume, Akademie-Verlag Berlin 1965.

[5] — Absolut p-summierende Abbildungen in normierten Bau/men, Studia Math.

28 (1967), pp. 333-353.

[ 6 ^] P. S a p h a r , Applications a puissance nucléaire et applications de Hilbert-Schmidt dans les espaces de Banach, Annales de l’ Ecole Normale Supérieure, t. 83 (1966), pp. 113-151.

[7] B . S ik o r s k i, Bemarks on Leèanski’s determinants, Studia Math. 20 (1961), pp. 145-161.

[ 8 ] — The determinant theory in Banach spaces, Coll. Math. 8 (1961), pp. 141-198.