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Chapter V
Nuclear Operators in Banach Spaces
This chapter contains a brief exposition of nuclear operators in Banach spaceand their corresponding trace and determinant. It also contains generalizationsto Banach space of some of the results in the preceding chapter. One of the maintheorems is Theorem 3.1 concerning the trace and determinant of nuclear operatorsdue to Grothendieck. Some asymptotic behavior of eigenvalues of nuclear operatorsin Banach spaces is presented.
In this chapter we assume that the Banach spaces B have the approximationproperty. Recall that a Banach space B has the approximation property if for everycompact K C B and every E > 0 there is an operator F of finite rank such thatIlx - Fxll < E for all x E K.
1 The Ruston-Grothendieck algebra of nuclear operators
Given a Banach space 8 with the approximation property, let D = D(8) consistof all operators A E 8 which admit representations
where
00
('{Jk E 8 , !k E 8') , (1.1)
Define
00
L II '{Jk IIBII fk IIBI< 00.
k=l
(1.2)
(1.3)00
II A lit = inf L II '{Jk IIBII fk IIBIk=l
The infimum in (1.3) is taken over all representations A in the form (1.1). SinceIIAxl1 -:; L~=l II '{Jk liB II fk IIBI Ilxll, it follows that
II A II£(B) -:; II A III . (1.4)I. Gohberg et al., Traces and Determinants of Linear Operators© Birkhäuser Verlag 2000
92 Chapter V. Nuclear Operators in Banach Spaces
It is easy to verify that \IAIIl is a norm in V. Operators A E V are called nuclear(or trace class) operators in Banach space B.
Theorem 1.1 The set V endowed with the norm IIAlll is an embedded subalgebraof £(B) with the approximation property.
Proof. For any operator A E V of the form (1.1) and any B E £(B), we have
00 00
II AB 111 ::::; II 2: 4'k 0 B' fk II ::::; II B IIL(B) 2: II 4'k 11811 ik 118/k=l k=l
and00 00
II BA 111 ::::; II 2: B4'k 0 fk II ::::; II B IIL(B) 2: II 4'k 11811 ik 118/k=l k=l
It follows from the above that V is a two-sided ideal in £(B). In particular, V isan algebra and
II AB 111 ::::; II A 111 II B IIL(8) ,II BA 111 ::::; II A 111 II B IIL(8) . (1.5)
From (1.4) and (1.5) we have that for any A, G in V,
II AG 111 ::::; II A Ihll G Ih . (1.6)
Hence V is an embedded subalgebra of £(B).The algebra V has the approximation property. Indeed, given A = L~l 4'k 0
fk , take Fn = L~=l 4'k 0 ik· Then
00
IIA - Fnll l ::::; 2: II4'k 11811 fk 118/ -t O.k=n+l
oThe nuclear space introduced above is complete. This was proved by Gro
thendieck. For details see ([Grl], Ch.II, Sec. 1), ([Gr2], Ch. I, Th. 1, Sec. 2 andProposition 35, Sec. 5). This will not be used in the sequel except in the proofs ofTheorems 2.2, 2.3.
Theorem 1.2 If the Banach space has the approximation property, then the functional trF is bounded on :F with respect to the norm IIF111. The extended trace ofoperator A = L~l 4'k 0 fk in V is given by
00
trv A = 2:(4'k,fk)k=l
and it does not depend on representation (1.1).
(1.7)
1 The Ruston-Grothendieck algebra of nuclear operators 93
In the particular case when B is a Banach space with basis, the proof of thistheorem is clear enough. Indeed, let {zn} and {Wn} be the basis and biorthogonalsystem of functionals respectively. Then the sequence
n
Kn = L Zm0Wmm=1
tends pointwise to identity I and IIKnIIL(B) ::; G. Let A be represented in the form(1.1). Then
and
n
tr AKn = tr L AZm 0 Wm
m=1
noon 00
= L (Azm,Wm) = L(L (ipk' Wm)Zm,fk) = L(Knipk' ik)m=1 k=1 m=1 k=1
00
Itr AKnl ::; GL IlipkllBllfkllBI.k=1
(1.8)
(1.9)
It follows from the estimate I(Knipk' fk)1 ::; GllipkllBllfkllB' and (1.2) that theseries in the right-hand side of (1.8) converges uniformly with respect to n. Since(Knipk' fk) -t (ipk' fk) for each k it follows from (1.8) and (1.9) that tr AKn -t
2::%"=1 (ipk' fk) for any representation (1.1), and that Itr AKnl ::; GIIAI11. Thus forany representation (1.1) we have:
00
IL(ipk, fk)1 ::; GIIAI11.k=1
(1.10)
In particular for any operator F of finite rank we obtain Itr FI ::; CI1P111. ThustrF admits a continuous extension from :F to D and for any representation (1.1)it follows that
n n 00
trv A = lim tr L ipk 0 fk = lim L(ipk, fk) = L (ipk' fk)' (1.11)k=1 k=1 k=1
The idea of the proof of the theorem in the general case (Banach algebras withthe approximation property) is to show that given A E D, there exists a generalizedsequence (net, d. [DSj, 1.7.1) which converges to 2::%"=1(ipk,fk) independent ofrepresentation (1.1).
Proof of Theorem 1.2. First we show that for any convergent series 2::%"=1 Ck ofpositive numbers, there exists a sequence {,6di'" of positive numbers such that
94 Chapter V. Nuclear Operators in Banach Spaces
{,Bd ---. 0 and 2:.':=1 (Ck /,Bk) < 00. Indeed, let 0 = ko < k1 < k2 < . .. be positiveintegers such that
00 1 00
L Ck::; 4n L Ck, n = 0, 1,2, ....k=kn +l k=1
(1.12)
Define 13k = 1/2n , kn < k ::; kn+1, n = 0,1,2, .... Then for C = 2:.':=1 Ck we havefrom (1.12),
00 k 1 k2 k3 1 1L (Ck /,Bk) = L Ck + L 2Ck + L 22Ck + ... ::; c(1 + 2+ 22 + ... ) = 2c.k=1 k=1 k=k 1 +1 k2 +1
For every F = 2:.7=1 'l/Jj ® gj in F, AF = 2:.7=1 A'l/Jj ® gj. Hence
n n 00
trAF= L{A'l/Jj,gj) = LL{'l/Jj,fk){'Pk,9j)j=1 j=lk=1
(1.13)
00 n 00
= L L{'l/Jj, fk){'Pk,gj) = L{F'Pk, ik)·k=lj=1 k=1
Since B has the approximation property, given any precompact K of B, there isan operator FK of finite rank such that
IIx - FKxl1 ::; 1 (1.14)
for all x E K. By what was shown above, there exists a sequence {,Bdk::l'.Bk > 0,such that 13k ---. 0 and
Now the set
f II'Pkllllikll < 00.
k=1 13k(1.15)
(1.16)
(1.17)
13k 00
K o := {-II-II 'Pd'Pk k=1
is precompact. Hence given E > 0 and precompact subset K ~ B with K ~
(I/E)Ko, we have (,Bk/EII'Pkll)'Pk E K and therefore
EII'Pk IIII'Pk - FK'Pkll ::;~
by (1.14). From (1.13), (1.17) and (1.15) we get
00 00 00 II'PkllllfkllIL{'Pk,ik)-trAFKI=IL{'Pk-FK'Pk,fk)I::;EL .Bk . (1.18)k=1 k=1 k=1
1 The Ruston-Grothendieck algebra of nuclear operators 95
The generalized sequence we seek is defined as follows: Partially order the collectionof all precompact subsets of B by K :S N if K ~ N. Define S K = tr AFK, whereK is precompact and FK is an operator of finite rank which satisfies inequality(1.14). Given any representation A = L:%"=l 'Pk@fk, the generalized sequence {SK}converges to L:%"=l ('Pk, ik)· For if K ::::: (I/E)Ko, then by (1.18)
Since generalized sequences in C have at most one limit, it follows that the seriesL:%"=l ('Pk, ik) is independent of the representation of A in (1.1).
The functional tr F is II . Ill-bounded on V. For suppose F = L:7=1 'l/Jj @gj'Then for any representation F = L:%"=l TJk @ hk,
n 00 00
tr F = IL('l/Jj,gj)1 = IL(TJk,hk)1 :S L IITJkllllhkll·j=l k=l k=l
Hence Itr FI :S IIFlll and therefore tr F admits a continuous extension trv(A) toV with
Itrv(A)1 :S IIAliI· (1.19)
Let A E V be represented in the form (1.1) and let An = L:~=l 'Pk @ fk. Then
n
trv(A) = lim tr An = lim L 'Pk @ fk.k=l
Thus
The theorem is proved.
00
trv(A) = L 'Pk @ ik·k=l
oFrom Theorem II.2.1 and II.3.2 we have that det(I + F) can be continuously
extended from F to V and for any A E V the function detv (I + AA) is an entirefunction of A. Moreover, the function detv(I + AA) is a Lipschitz function on anybounded subset on V. This follows from the following theorem.
Theorem 1.3 For any operator A E V(B) the following estimates hold:
I detv(I + A) I :S (1+ II A IiI) exp(2 II A Iii)
and for any two operators A, B E V
I detv(I + A) - detv(I + B) I:SII A - B IiI (2+ II A III + II B lid exp(2(1+ II A III + II B Ild2
) .
(1.20)
(1.21)
96 Chapter V. Nuclear Operators in Banach Spaces
Proof. Using the continuity argument it is enough to establish estimates (1.20)and (1.21) for operators of finite rank. Let F = 2:::=1 'Pm 0 1m. Then by 1.5.3
m 1det(I + F) = 1 + 2: -,
m.m=l
N
2:il,··"im=l
Since I ('Pip,fiq) I :S II 'Pip IIBII Jiq IIB" it follows from Hadamard's inequalitythat
m
Idet(('Pip ,liq ));'q=ll:S m T II II 'Pip IIBII Jiq liB/p=l
Hence
N
2: det (('Pip, Jiq));'Q=li l , ... ,im =l
and
N m
< m T 2: II II 'Pip IIBII lip liB/i 1 , ... ,im =1 p=l
m m (N )mdet(I + F) :S 1 + 1; :~ ~ II 'Pi IIBII Ii liB/ .
Taking the infimum in the right-hand side of this inequality we obtain
m m
~m2Idet(I + F) I :S 1 + LJ -, II F II~:S (1+ II F lid exp(2 II F Iii) .m.
m=l
(1.22)
(1.23)
The last inequality may be obtained as follows:Let x =11 Fill and let bm = mm/21m!, m = 1,2, .... It is readily seen that
b2m+l :S b2m :S 2m1m!. Hence
00 00
2: b2m X2m + 2: b2m+lX2m+1
m=l m=l
00 00 2m
< (1 + x) 2: b2m X2m :S (1 + x) 2: -,x2m
m=l m=l m.
< (1 + x)(e2X2- 1) :S (1 + x)e2X2
- 1
which establishes the last inequality in (1.22). Applying Corollary 4.2, Ch. II andthe estimate (1.22), we obtain (1.20) and (1.21).
Note that for Hilbert space 1t, the Ruston-Grothendieck algebra V = V(1t)coincides with the algebra of trace class operators. In order to prove this statement
2 Examples of nuclear operators in Banach spaces 97
it is enough to show that for all operators F E F the norm II F 11'0 coincides withthe trace norm II Fill' It is evident that for operators <P 0 f of rank 1
(1.24)
NLet F E F. Then for any representation F = 2: <Pm 0 fm we have
m=l
m m
II F Ih:s L II <p0fm 111= L II <Pm IIBII fm IIBIm=l m=l
and hence
II F 111:S11 F 11'0 . (1.25)
NLet F = 2: sj(F)<pj 0'l/Jj be the Schmidt decomposition of operator F. Here
j=lsj(F) are the singular numbers 0 f F and {<pj}, {'ljJj} are orthonormal systems.Then
n
II F II'D:S L sj(F) =11 Fillj=l
Taking into account inequality (1.25) we obtain
II F 11'0=11 Fill . (1.26)
Note that the notion of nuclear operators in Banach spaces given above isnot unique. There are more generalizations of the notion of trace class operatorsfrom Hilbert to Banach spaces. For details see the books [Pi1] and [Ko4] and thecomments to this chapter. In this book we restrict ourselves to the Ruston-Grothendieck definition of nuclear operators.
2 Examples of nuclear operators in Banach spaces
In this section we give some examples of algebras of trace class operators in Banachspaces £1, era, b], L1(T, 2:,j.L).
Theorem 2.1 Let !3 = £1 and let operator A be defined by matrix A = (ajk)rk=lwith respect to the standard basis in £1. Then operator A is nuclear in £1 if andonly if
00
LSup Iakm 1< 00 .
k=l m
Let A be a nuclear operator. Then00
(2.1)
(2.2)
98 Chapter V. Nuclear Operators in Banach Spaces
Proof. Let {edi'" be the standard basis in £1. Denote by rpm the sequences rpm ={amj}~l = A'em, then
00
Suppose that (2.1) is fulfilled. Then
00 00
IIAlll :S L II ek 11811 rpk 118'= LSup Iakm 1< 00
k=l k=l m
(2.3)
and hence A is nuclear in £1. Conversely, let A be a nuclear operator in £1. Thenthere exists a representation
00
(2.4)
such that 'l/Jm E £1, 1m E £00 and
00
L II 'l/Jm IIe1 11 1m Ileoo < 00 .
m=l
It can be directly checked that
00
A'ek = L('l/Jm,ek)lm andm=l
and hence
00
II Wm Ile l = L I ('l/Jm, ek) Ik=l
00
LSup Iakm Ik=l m
00
L II A'ek Ileoo
k=l00 00 00
(2.5)
< L L 111m Ileoo I ('l/Jm, ek) 1= L 111m Ileoo II 'l/Jm Ile l < 00 .
k=l m=l m=l
This estimate is true for any representation of the operator A in the form(2.4) and hence
00
LSup Iakm I:S IIAIIt· (2.6)k=l m
Equality (2.2) follows from (2.3) and (2.6). The theorem is proved. 0
Note that the algebra of nuclear operators acting in £1 coincides with thealgebra D(£l) considered in Section III.5 (and the notations coincide!).
2 Examples of nuclear operators in Banach spaces 99
Theorem 2.2 Let k(t,s) be a continuous function on a square [a,b] x [a,b]. Thenthe integral operator
(Af)(t) = l b
k(t, s)f(s)ds
is nuclear in C[a, b].
Proof. For any n E N there exists a On such that
(2.7)
(2.8)
(2.9)
for ISl - S2 I< On and t E [a, b]. Let Rn be a following partition of the segment [a,b]:
Rn = {sndf=l (N = N(n);a = SnO < Sn1 < ... < SnN = b;Sn,k+l - Sn,k < on).
We assume that Rm J Rn for m > n. Denote by An the integral operator withkernel
N-1kn(t, s) = L k(t, snp) X[Snp,Sn,p+l](S) ,
p=O
where XM is the characteristic function of the set M. Operator An has a representation
N-1An = L 'l/Jpn @ fpn
p=o(2.10)
where 'l/Jpn(t) = k(t, snp) and
fpn(<P) = jSn,p+l <p(s)ds . (2.11)Sn,p
We show that {An} is a Cauchy sequence in the algebra V(C[a, bJ). Let m > n,then
N(m)-l Sm,p+l
(Am - An)<p(t) = L (k(t, Smp) - k(t, Snj)) j <p(s)ds ,p=o Sm,p
where I smp - Snj I< On' Hence
1 N(m)-l 1 N(m)-l b - aII Am - An III ~ ;;: L II fpm II = ;;: L (Sp+l,m - Spm) = ---;;-.
p=O p=O
Since V(C[a, b]) is a Banach space,there exists an operator B E V(C[a, b]) suchthat II An - B 111-; O. It follows from (1.9) that II An - B IIL(B)-; O. But it can bereadily checked that II An - A IIL(B) ~ ~ -; 0, hence A = B, i.e., A is a nuclearoperator in C[a, b]. 0
100 Chapter V. Nuclear Operators in Banach Spaces
Let T be a compact set and let J.L be a non-negative measure on an algebraE of subsets of T. Denote by L1,00(T, E, J.L) a Banach space of all measurablefunctions f(t, s) with the norm
II f IIL1 ,00 = r ess sup I f(t, s) Ids .iT tET
(2.12)
By L1,00(T, E, J.L) we denote the closure with respect to the norm (2.12) ofthe set S of the functions
N
a(t,s) = Lfk(t)CPk(S) (CPk E L1(T), fk E Loo(T)). (2.13)k=l
Note that L1,00 =f L 1,00. For example one can take T = [0,1] with standardLebesgue measure and the function
f(t s) = {I , 0:<::; t :<::; s :<::; 1, 0, O:<::;s<t:<::;l.
It is not difficult to prove that f E L1 ,00 but f ~ L1,00.
Theorem 2.3 Let T be a compact set and let J.L be a non-negative measure on analgebra E of subsets ofT. Suppose that k(t, s) is a measurable function on TxT.Then the integral operator
Acp(t) = Lk(t, s)cp(s)ds
is nuclear in L1(T,E,J.L) if and only if k E L1,00(T,E,J.L).Let A be nuclear in B = L1(T, E, J.L). Then
II A IIv(B)= resssup I k(t, s) IdJ.L(s).iT tET
Proof. Denote by So the set of all functions of the form
N
g(t, s) = L ak(t)xk(S) ,k=l
(2.14)
(2.15)
(2.16)
where ak E Loo(T) and Xk are characteristic functions of disjoint measurablesubsets Tk cT. Let a function a(t, s) E S be represented in the form (2.13). Forany CPk E L1(T) and any E > 0 there exists a simple function CPk which takes afinite number of values, such that II CPk - CPk II L 1 (T) < E. Let us consider the followingfunction
N
7i(t, s) = L h(t)CPk(S).k=l
(2.17)
2 Examples of nuclear operators in Banach spaces
It is clear that aE So and
N
Iia - a11L1 ,00 < EL IlfkllLoo '
k=l
101
(2.18)
It follows from here that the set So is dense in £1,00 in the norm (2.15). In particularthe kernel k(t, s) of the operator A can be approximated in the L1,00 norm by thefunctions kn(t,s) E So.
Let F be an integral operator with kernel g, defined by (2.16). Then for thenuclear norm of F we have:
N
II F III ~ L II ak IILoo II Xk IIL1
k=l(2.19)
N
= L esssup lak(t)!JL(Tk) = ( esssup 1g(t, s) IdJL(s) =11 9 111,00 .k=l tET iT tET
Denote by {Kn} the sequence of integral operators with kernels kn(t,s) E So.Since
Ilk(t,s) - kn (t,s)IIL1 •00 -t 0, (2.20)
it follows from (2.19) that {Kn } is a Cauchy sequence in the Banach algebra ofnuclear operators and it tends to a nuclear operator B (in both algebras V(L1 )
and L(L1)). It can be easily deduced from (2.20), that K n tends to A in V(L1),
hence B = A. Thus A is a nuclear operator. Conversely, let the integral operatorA with kernel k(t, s) be nuclear in L1(T, 'L., JL). It follows from the definition ofnuclear operators that for a given E > 0 there exists a representation
00
A = 'LJi 0 aii=l
such that
f 11 fi(S) IdJL(s):s (1 +E) II A IIIi=l T
It follows from here that the function00
f(t,s):= LJi(s)ai(t)i=l
is defined almost everywhere on T and
(2.21)
(2.22)
£ess ~~? If(t, s) IdJL(s) :S~ £1 Ji(s) IdJL(s) :S (1 + E) II A III (2.23)
102 Chapter V. Nuclear Operators in Banach Spaces
Thus f(t, s) E £1,00' It follows from (2.21), (2.22) that f(t, s) is a kernel of theintegral operator A. Since k(t, s) is also a kernel of A, k(t, s) = f(t, s) a.e. on Tand hence k(t,s) E £1,00'
Equality (2.15) follows from (2.19) and (2.23). 0
3 Grothendieck trace theorem
Let A be a nuclear operator in Banach space B. If B is a Hilbert space, then thefollowing equalities (Lidskii Theorem IV. 6.1) hold:
tr A = L Aj(A), det(I + A) = II(1 + Aj(A)) ,j j
(3.1)
where {Aj(A)} is the set of all eigenvalues of operator A, multiplicities taken intoaccount. The following example shows that there exist nuclear operators in B forwhich (3.1) is not true.
There exists an even function ¢(t) continuous on [-7f,7f] such that the corresponding Fourier series
N
¢(t) rv L Cn cosnti=l
(3.2)
diverges at the point t = O. One can take such an example constructed by Lebesgue(see in [B], Ch. 1, § 46). Consider in the space C[-7f,7f] the integral operator
A<p(t) = i: ¢(t - s)<p(s)ds .
It follows from Theorem 2.2 that A is a nuclear operator in C[-7f,7f]. It is wellknown ([Ho], Ch.1) that the set {Aj (A)} of all eigenvalues of operator A coincideswith the set {Cn } of all Fourier coefficients of the function ¢. Since the Fourierseries in (3.2) diverges at the point t = 0, the series
(3.3)
also diverges.Note that in this example the product
(3.4)
also diverges (see Corollary 4.2 below).
3 Grothendieck trace theorem 103
A question arises: maybe it is always possible to save the Lidskii Theorem fornuclear operators in Banach spaces by appropriate renumeration of the eigenvalues? The answer is no! P. Enflo [E] constructed an example of a nuclear operatorA in £1 such that
and trv A = 1
and for which it is impossible to renumerate Aj(A) = 0 to obtain Lj Aj(A) = I!Some simplified constructions of this example are given in [LT] Vol. I, Theorem2.d.3 and [Pi1]' 10.4.5.
The following Grothendieck Theorem shows that the Lidskii Theorem is validfor a subclass of nuclear operators - the so-called ~-nuclear operators.
Theorem 3.1 Let A E L(8) and suppose that A admits a representation
00
where C{Jk E 8, ik E 8' and
00
L II C{Jk 112
/3
11 ik 112
/3< 00 .
k=l
(3.5)
(3.6)
Let {Aj (A)} be the set of all eigenvalues of A with multiplicities taken intoaccount. Then
and the following equalities hold:
trv A = L Aj(A) ,j
detD(I + A) = IT (1 + Aj(A)) .j
(3.7)
(3.8)
(3.9)
Proof. It follows from (3.6) that A is a nuclear operator in 8 and II A - An IiI -70,n
where An = L C{Jk (8) ik·k
By the definition of trv A and detD(I+A) in the algebra D = D(8) of nuclearoperators we have
trv A = lim tr An , detD(I + A) = lim det(I + An). (3.10)n~CX) n-+oo
104
In Chapter I we showed that
Chapter V. Nuclear Operators in Banach Spaces
and
n
tr An = L)'Pk, Ik)k=1
(3.11)
(3.12)
Consider two infinite matrices:
(3.13)
where f3n =11 'Pn 11 1/
3 11 In 11 1/
3 and
(3.14)
withII r 11 1
/3 11 'Pk 11
1/3
bjk = II;j 11 2/311 Ik 112/3 ('Pj,lk) .
It follows from condition (3.6), that
00 00
L I f3m 12< 00 and L I bjk 1
2< 00 .
m=1 j,k=1
Thus the infinite matrices C and B can be considered as Hilbert-Schmidt operatorsin Hilbert space £2. Denote M := CB. Operator M is oftrace class in Hilbert space£2 and its matrix representation in standard basis is
(3.15)
Let
Since M is of trace class (in £2) we have
(3.16)
trMn
lim tr Mn = lim ~ ('Pk, Ik)n~oo n---too ~
k=1lim tr An = trv A
n--oo
(3.17)
3 Grothendieck trace theorem
and
105
det(I + M) lim det(I + Mn ) (3.18)n->oo
. ( II fj 112
/3
II 'Pk 111
/3
)nl~det 8jk + II 'Pj 11 1/ 3 ('Pj,!k) 11!k 11 2/ 3
lim det(8jk + ('Pj,!k))n->oolim det(I + An) = detD(I + A).n->oo
The same equality is valid for operator >'A, i.e.,
detD(I + >'A) = det(I + >'M) . (3.19)
The following important statement is a consequence of equality (3.19) and Theorem 11.6.2. The set of eigenvalues {A(A)} of operator A in Banach space B (themultiplicities are taken into account) coincides with the set {>'(M)} of eigenvaluesof operator M in £2. Using the Lidskii Theorem IV.6.1 we obtain
j j
anddetD(I + A) = det(I + M) = II(1 + >'j(M)) = II(1 + >'j(A))
j j
and the theorem is proved. D
Corollary 3.2 Let L(>.) = 1+ >'A1 + ... + >.nAn' and suppose that the operatorsAm (m = 1, ... , n) admit representations
Am = AmI·· ·Amm , (3.20)
where Amj(j = 1, ... , m) are ~-nuclear operators (i.e., satisfy the condition ofTheorem 3.1). Let /L1l /L2, ... be all characteristic numbers of the pencil L(>') (multiplicities are taken into account). Then
12:-1 -I <00 ,n /Ln
12:- = -trA In /Ln
(3.21 )
and
II (1-~) = detL(>') .n /Ln
This statement can be deduced from Theorem 3.1 using the same arguments as inthe proof of Theorem IV. 10.1.
The definition of characteristic numbers of the pencil L(>') appears in Section10, Ch. IV.
106 Chapter V. Nuclear Operators in Banach Spaces
4 Asymptotic behavior of eigenvalues of nuclear operators
Let A be a nuclear operator in Banach space 13 with {Aj (A)} the set of all eigenvalues of operator A (multiplicities taken into account). If 13 is a Hilbert space,then
(4.1)
In the previous section we gave an example of a nuclear operator in era, b] forwhich
L I Aj(A) 1= 00.
j
Moreover, there exists an integral operator with continuous kernel such that
(4.2)
for any p > 2, (see T. Carleman's example Ch. IV, Section 8). It turns out thatp < 2 is a sharp result. Namely, the following theorem holds.
Theorem 4.1 Let A be a nuclear operator in a Banach space 13, then
(4.3)
where {Aj(A)} is the set of all eigenvalues of operator A, taking into account themultiplicities.
Proof. By the definition of nuclear operators in Banach spaces, there exists arepresentation
00
A = L'Pk Q9!kk=l
of operator A such that 'Pk E 13, fk E 13' and
00
L II 'Pk liB II fk IIB'< 00 .
k=l
One can suppose that 'Pk f. 0, fk f. 0. Denote by M an infinite matrix
where
(4.4)
(4.5)
(4.6)
4 Asymptotic behavior of eigenvalues of nuclear operators
Since
107
00 00
L I ajk 12
::; L II Ii 11II rpj 1111 Ik 11I1 rpk 11< 00, (4.7)~k=l ~k=l
the matrix M can be considered as a Hilbert-Schmidt operator in Hilbert space£2 (with respect to the standard basis in £2)'
Denote by An the operator of finite rank in 13:
and by M n the finite matrix
Mn = (ajk)7,k=l .
Since A is a trace class operator in 13 we have:
(4.8)
(4.9)
trv A = lim tr An and detD(I +A) = lim det(I + An). (4.10)n---+oo n-+oo
We know (see Theorem I. 3.1) that
n
tr An = L (rpm, 1m) = tr Mnm=l
and
(4.11)
(4.13)
det(I + An) = det (Ojk + (rpj, Ik)) = det(I + Mn). (4.12)
It follows from the definition of det2(I + F) (see Ch. IV, Section 7) that
det2(I + Mn) = det(I + Mn)exp( - tr Mn)= det(I + An) exp( - tr An) .
Since M is a Hilbert-Schmidt operator there exists a limit in the left-hand side of(4.13) and
det2(I + M) = detD(I + A) exp( - trv A) .
The same equality holds for operator AA:
det2(I + AM) = detD(I + AA) exp(-A trv A) . (4.14)
Thus the entire functions deh(I + AM) and detD(I + AA) have the same sets ofzeros, taking into account their multiplicities. It follows from here and Theorems11.6.2, that the set {Aj(A)} and {Aj(M)} of eigenvalues of operators A and M in13 and £2 respectively coincide. Hence
L I Aj(A) 12= L I Aj(M) 1
2 < 00. (4.15)j j
The last estimate follows from (3.3) Chapter IV. The theorem is proved. 0
108 Chapter V. Nuclear Operators in Banach Spaces
The next statement generalizes both inequalities (3.7) and (4.3).
Theorem 4.2 Let operator A admit a representation
00
and
00
(!Pn E [3, fn E [3') (4.16)
L II !Pn 11811 fn 118'< 00
i=l
for some r (0 < r :S 1). Then
(4.17)
00
L I An(A) IP< 00
n=l(4.18)
Proof. It follows from (4.17) that A is a nuclear operator, hence (see the proof ofTheorem 4.1) the set {Aj(A)} coincides with the set {Aj(M)} of eigenvalues ofoperator M = (ajk)rk=l in £2, where
(4.19)
Operator M can be presented as a product
where M 1 = diag(b1 , b2 , ..• , bn , ... ) with
bn = (II 'Pn II . II fn 11)(1-r)/2
and M 2 = (GJm)OO -1 with},m-
It follows from (4.17), (4.19) and (4.21) that
(4.20)
(4.21)
(4.22)
00 00
L I Gjm 12 :s L (II !Pj I1II fj 11I1 !Pm 1I11 fm IIr < 00 .
j,m=l j,m=l
Hence M 2 is a Hilbert-Schmidt operator, and therefore
00
Ls;(M2 ) < 00.
j=l
5 Comments 109
It follows also from the definition of M1 and condition (4.17) that for q = 2r/(l-r)
00 00
LSJ(Md = L(II 'Pn 11·11 fn Ilr < 00.
j=l j=l
Since ~ + ~ + ~ = ~ - ~ = ~ it follows from Theorem IV.11.2 that
00 00
Ls;(M) = Ls;(M1M2M1) < 00.
j=l j=l
Thus00 00 00
L IAj(A) IP= L IAj(M) IP~ Ls;(M) < 00.
j=l j=l j=l
In the last equality we used the estimate (3.3) from Chapter IV. o
Corollary 4.3 Let A be a nuclear operator in a Banach space and let {Aj} be theset of all eigenvalues of operator A, taking into account multiplicities. Then thesenes
converges if and only if the product
converges.This statement follows from (4.3) and the following known exercise from
calculus:00 00
Let {en} E £2· Then the series L en converges ifand only if the product IT (1+en)n=l n=l
converges.
5 Comments
The notion of nuclear operators in Banach spaces was introduced independentlyby Ruston [Ru1] and Grothendieck [Gr1]. Here the existence and analyticity ofthe determinant in the Ruston-Grothendieck algebra we obtain as consequencesof general Theorems II.2.1 and 11.3.1. The proof of Theorem 1.2 follows the proofof Lemma 4.7.1 in [Pi1]. Theorem 1.3 seems to appear for the first time. The examples considered in Section 2 are classical (we used Grothendieck's paper andPietsch's book). Grothendieck's trace theorem for ~-nuclear operators was obtained in ([Gr2], Chap. II, page 19). Here (in Section 3) we proved this theoremby reduction to Lidskii's trace theorem and (modulo the Lidskii Theorem) this
110 Chapter V. Nuclear Operators in Banach Spaces
proof looks elementary. For the history of the asymptotic behavior of eigenvaluesof nuclear operators (Theorems 4.1 and 4.2) we refer to ([Pi1]' page 298). Here weobtain these results by reduction to Hilbert-Schmidt operators.
In this book we restricted ourselves to the Ruston-Grothendieck generalization of the trace class operators to Banach spaces. An interesting concept of"operators with trace" in Banach spaces was given by A. Pietsch (see [Pi2]). Heconsidered a class Ql of operators A defined by the condition {/lm (A)} E h, where/lm (A) is the distance from A to the set of all operators of rank not exceeding n -1.Pietsch proved that in any Banach space, the trace can be extended to all operators from Ql. In the paper [Ko1] Hermann Konig showed that for operators fromQl, the following analogue of the Lidskii theorem holds, namely,
00 00
L /In(A) = L An(A)n=1 n=1
where {Aj(A)} is the set of all eigenvalues of operator A, taking into account themultiplicities. Note that for a smaller class of operators this equality was obtainedearlier by A. Markus and V. Matsaev [MM], who assumed that
00
L /In(A) In(/l~I(A) + 1) < 00.
n=1
For this and other concepts of nuclear operators in Banach spaces we referthe reader to the books [Pi1] and [Ko4].
