An alternative to Plemelj-Smithies formulas on infinite determinants

18 Pages·0.14 MB·English

by  Domingos H. U. Marchetti

AN ALTERNATIVE TO
PLEMELJ-SMITHIES FORMULAS ON INFINITE
DETERMINANTS
Domingos H. U. Marchetti
3
Department of Mathematics and Statistics
9
9 McMaster University
1
Hamilton, Ontario L8S 4K1, Canada

n
a
J

1
1

1
v
2
0
0 ABSTRACT: An alternative to Plemelj - Smithies formulas for the p -regularized quan-
1
tities d(p)(K) and D(p)(K) is presented which generalizes previous expressions with p = 1
0
3
due to Grothendieck and Fredholm. It is also presented global upper bounds for these
9
/ quantities. In particular we prove that
n
y
-d |d(p)(K)| ≤ eκkKkpp
o
a
h holdswithκ = κ(p) ≤ κ(∞) = exp − 1 forp ≥ 3whichimprovespreviousestimate
c 4(1+e2π)
: yielding κ(p) = e(2+ln(p−1)).
v (cid:8) (cid:9)
i
X
r to appear in J. Funct. Analysis
a
1
1. INTRODUCTION
The Fredholm Theory[1], as addressed in this paper, is concerned with the problem of
solving the equation
(1+µ K)f = f (1.1)
0
in a separable Hilbert space H with K belonging to the trace - class of operators on H or,
more generally[2,3,4,5], K is considered to be a compact operator of the class C = {A :
p
kAkp ≡ Tr(|A|p) < ∞}. Fredholm theory has been mainly applied in Scattering Theory
p
but (1.1) also appears in a variety of problems in Many Body Theory and Quantum Field
Theory (see Simon[6,7] for a review and applications).
ThebasicresultoftheFredholmTheoryistowritetheresolventR(K;µ) ≡ (1+µK)−1
of the operator K as a quotient
D(p)(K;µ)
R(K;µ) = , p = 1,2,... (1.2)
d(p)(K;µ)
of entire functions of µ. Therefore, (1.1) has a unique solution for any operator of the class
C , given by
p
f = R(K;µ) f (1.3)
0
provided −1/µ is not an eigenvalue of K.
d(p) and D(p) can be explicitly written in terms of their power series
µn
d(p)(K;µ) = d(p)(K) (1.4)
n! n
n≥0
X
µn
D(p)(K;µ) = D(p)(K) (1.5)
n! n
n≥0
X
(p) (p)
where d are complex valued and D are operator valued coefficients given by the
n n
Plemelj[3] - Smithies[4] formulas
σ n−1 0 ... 0
1
σ σ n−2 ... 0
2 1
(cid:12) (cid:12)
d(p)(K) = (cid:12)σ3 σ2 σ1 ... 0 (cid:12) (1.6)
n (cid:12) . . . . . (cid:12)
(cid:12) .. .. .. .. .. (cid:12)
(cid:12) (cid:12)
(cid:12)σ σ σ ... σ (cid:12)
(cid:12) n n−1 n−2 1(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
and (cid:12) (cid:12)
K0 n 0 ... 0
K1 σ n−1 ... 0
1
D(p)(K) = (cid:12)(cid:12)K2 σ2 σ1 ... 0 (cid:12)(cid:12) (1.7)
n (cid:12) . . . . . (cid:12)
(cid:12) .. .. .. .. .. (cid:12)
(cid:12) (cid:12)
(cid:12)Kn σ σ ... σ (cid:12)
(cid:12) n n−1 1(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) 2 (cid:12)
with K0 ≡ I and
Tr(Kj) if j ≥ p
σ = σ (p) = (1.8)
j j
0 otherwise
(cid:26)
(p)
Expression(1.5)istobeinterpretedinthesense that(φ,D (K)ψ)isthedeterminant
n
of the matrix (1.7) with the operator Ki replaced by (ϕ,Ki ψ), for ϕ,ψ ∈ H.
Thefunction d(p) iscalledthep-regularizeddeterminant andwewritehere Poincar´e’s
definition[2]
∞ µj
d(p)(K;µ) = det (1+µ K) ≡ exp (−1)j+1 Tr (Kj) (1.9)
p
j
(cid:26)j=p (cid:27)
X
Although definition (1.9) seems to require µkKk < 1 (see Simon[6] for other defini-
p
tion), it can be used to derive (1.6) for µsufficiently small and by Hadamard’s inequality[4],
or by a limite procedure[5], one can prove analyticity of (1.4) for all µ ∈ C.
In this note (1.9) is used to derive an alternative formula for d(p) and D(p), with
p = 1,2,..., which generalize the algebraic formula for the determinant d(1) due to
Grothendieck[9]. We then discuss the analytic properties of these series.
Grothendieck’s formula is given by the power series (1.4) where the n-th coefficient
d(1) = n! Tr(∧nK) ≡ T (∧nK) (1.10)
n n
is a trace of a n-fold antisymmetric tensor product of operators on H and, as recognized
by Simon[6], it has the following advantage: once one uses the simple bound
|T (∧nA)| ≤ kAkn (1.11)
n 1
the analytic properties of d(1) are established avoiding to use Hadamard’s inequality. It
is also worth of mention that (1.10) reduces to the definition of Fredholm[1] if K is an
integral operator with continuous kernel.
We shall state our results.
Theorem 1.1. Let K ∈ C . Then the power series (1.4) and (1.5) converge for all µ ∈ C
p
and their coefficients can be written as
s
d(p) = C(p) T (K|P1| ∧...∧K|Ps|) (1.12)
n " |Pi|# s
PX∈Pn iY=1
s
1
D(p) = (−1)|P1|−1|P |! C(p) K|P1|−1 T (K|P2| ∧...∧K|Ps|)
n n+1 P∈XPn+1 1 "iY=2 |Pi|# s−1
(1.13)
3
where P is the collection of partitions P = (P ,...,P ) of {1,2,...,n} and
n 1 s
k
d
(p)
C = +ξ (1+λ) (1.14)
k dλ p
" #
(cid:18) (cid:19) λ=0
with ξ = ξ (λ) such that ξ = 0 and for p ≥ 2
p p 1
p−2
ξ = (−1)j+1 λj (1.15)
p
j=0
X
Remark 1.2. One can check from (1.14) that
(p)
C = 0 for k = 1,...,p−1. (1.16)
k
Therefore (1.12) and (1.13) are well defined expressions since
|T (Kn1 ∧···∧Kns)| ≤ kKkni (1.17)
s ni
i
Y
is finite provided n ,...,n ≥ p. So, the effect of (1.16) in (1.12) is analogous to that of
1 s
(1.8) in the Plemelj - Smithies formula of d(p) (recall that |σ (p)|1/q ≤ kKk ≤ kKk for
n q q p
any q ≥ p).
Remark 1.3. Analyticity of (1.4) and (1.5) can be established by estimating (1.12) and
(1.13). If the following crude bound for (1.14)
p−2
|C(p)| ≤ 2 pk−1 k ! (1.18)
k p−1
(cid:18) (cid:19)
is used one can show covergence of these series for all µ ∈ C. In (1.18) 2pk−1 accounts
for the number of terms after expanding (1.14) and we then take the worse of these. It
is not difficult to provide an upper bound on C(p) which replaces 2pk−1 in (1.18) by a p
k
independent constant c.
A different expression for (1.9) is needed in order to obtain further analytic properties.
By Lidskii’s theorem[13,6] (1.9) can be written as
d(p)(K;µ) = E(−µγ ;p−1) (1.19)
i
i∈I
Y
where E(z;q) is the Weierstrass primary factor defined by
E(z;0) = 1−z (1.20)
4
and
q
zj
E(z;q) = (1−z) exp (1.21)
j
(cid:26)j=1 (cid:27)
X
for q > 0. Here {γ } is the collection of all eigenvalues of K, counted up to algebraic
i i∈I
multiplicity.
By an improved estimate on the Weierstrass factor E (Lemma 3.1.) which sharpens
previous bound on its type[12], we are led to the following result:
Theorem 1.4. Given p ≥ 1, let K ∈ C . The following inequalities hold
p
|d(p)(K;1)| ≤ eκkKkpp (1.22)
and
kD(p)(K;1)k ≤ 2 eκ(1+kKkp)p (1.23)
∞
with κ = κ(p) such that κ(1) = 1 , κ(2) = 1 and for p ≥ 3
2
−1 p−1 −1
p−1 p−2 π
κ = exp − 1+ 1+2 1+cosec . (1.24)
p 4p p
(cid:26) (cid:20) (cid:18) (cid:18) (cid:19) (cid:19) (cid:21) (cid:27)
Remark 1.5. The proof of Theorem 1.4 is straightforward for p = 1; Smithies[4] estab-
lished the result for p = 2 and for p = 4 Brascamp[5] obtained κ = 3. Our proof of theorem
4
−1
1.4 extends for arbitrary p the proof of ref. [5]. Notice that κ(p) ≤ κ(∞) = e4(1+e2π) is in
contrast with the previous estimate[12,6] resulting in κ(p) = e(2+ln(p−1)).
Theorem 1.1 and theorem 1.4 will be proved in section 3. Theorem 1.1 is based on
an explicit algebraic computation of the n-th derivative of d(p) (Lemma 2.3 of section
2). The expression derived in lemma 2.3 is important for organizing terms in the cluster
expansion of fermionic Quantum Field Theories[10]. Applications on this field will appear
elsewhere[11]. In section 4 it is presented a simple derivation of Fredholm’s formula and its
generalization.
5
2. THE BASIC LEMMA
We begin with a brief review on the antisymmetric tensor product. We use the
notation and some results of ref. [10].
Let ⊗nH = H⊗···⊗H be the n - fold tensor product of a Hilbert space H and let
∧nH = H∧···∧H donote its antisymmetric subspace. A ”simple” vector Φ ∈ ∧nH is of
the form
1
Φ = (−1)|π|ϕ ⊗ϕ ⊗···⊗ϕ
π(1) π(2) π(n)
n!
π
X
≡ Π(ϕ ⊗···⊗ϕ ) (2.1)
1 n
for some ϕ ,...,ϕ ∈ H. Here we sum over all permutations π = (π(1),...,π(n)) of
1 n
{1,2,...,n} and |π| counts the number of permutations required to return to the original
order. Π stands for the projection of ⊗nH into ∧nH. We write Φ = ϕ ∧···∧ϕ .
1 n
If {ϕ } is an orthonormal basis for H then {ϕ ∧ϕ ∧···∧ϕ } with i < i < ··· < i
i i1 i2 ir 1 2 r
isanorthonormalbasisfor∧rH, r = 1,2,.... Fromthis(1.11)anditsgeneralization(1.17)
can be proved.
If Φ , Ψ ∈ ∧nH are ”simple” vectors, their scalar product is given by the determinant
of a n×n matrix
1
(Φ , Ψ) = det{(φ ,ψ )} (2.2)
i j
n!
whose elements are scalar products in H.
Given K ,...,K bounded operators on H and Φ ∈ ∧nH we define
1 n
1
(K ∧···∧K )Φ = K ϕ ∧···∧K ϕ (2.3)
1 n π(1) 1 π(n) n
n!
π
X
i.e. K ∧···∧K = Π(K ⊗···⊗K )Π. We write ∧nK = K ∧···∧K.
1 n 1 n
IfK andLareoperatorson∧nHand∧mH,respectively, thenK∧L = Π(K⊗L)Πisan
operator in ∧n+mH. Moreover the product ∧ is commutative, associative and distributive
with respect to addition.
Given a bounded operator K on H, we define its derivation d(∧nK) on ∧nH by
d(∧nK) = n(K ∧I ∧···∧I) (2.4)
Lemma 2.1. Let K ,...,K and L be bounded operators on H. Then
1 n
∧nL·K ∧···∧K = LK ∧···∧LK (2.5)
1 n 1 n
n
d∧n L·K ∧···∧K = K ∧···∧LK ∧···∧K (2.6)
1 n 1 i n
i=1
X
6
proof: Appendix of [10].
Lemma 2.2[10]. Let A ,...,A be trace class operators on H. Then
1 k+1
T (A ∧···∧A ) = T (A )T (A ∧···∧A )−T (d∧k A ·A ∧···∧A ) (2.7)
k+1 1 k+1 1 k+1 k 1 k k k+1 1 k
proof: We notice that (1.10) implies
k+1
d
det(1+A(λ)) = T (A ∧···∧A ) (2.8)
k+1 1 k+1
dλ
" i! #
iY=1 λ=0
where A(λ) = λ A +···+λ A . We write
1 1 k+1 k+1
˜
det(1+A(λ)) = det(1+λ A )det(1+R A(λ)) (2.9)
k+1 k+1 k+1
˜
where R = R(A ;λ ) and λ = (λ ,...,λ ).
k+1 k+1 k+1 1 k
¿From (2.5), (2.8) and (2.9) we have
k
d
det(1+A(λ)) = det(1+λ A )T (∧kR ·A ∧···∧A ) (2.10)
k+1 k+1 k k+1 1 k
dλ
" i! #
iY=1 λ˜=0
We deduce (2.7) by differentiating (2.10) with respect to λ and setting λ = 0.
k+1 k+1
We are now ready to state our basic lemma.
Lemma 2.3. Let A ∈ C be a multivariable function and let D ,...,D be derivatives.
p 1 n
Then we have
n
D d(p)(A;1) = T (∧sR·A ∧···∧A ) d(p)(A;1) (2.11)
j s P1 Ps
(cid:18)jY=1 (cid:19) PX∈Pn
where R = R(A;1) = (1+A)−1 and for any subset Q of {1,...,n}
A = (D +ξ D A) (1+A) (2.12)
Q j p j
(cid:18)j∈Q (cid:19)
Y
with ξ = ξ (A) as in (1.15) with λ replaced by A.
p p
Remark 2.4. The cancelation leading to (1.16) occurs for (2.12): terms in the expansion
of R A are of the form
Q
AqD A...D A
Q1 Qn
7
for a q ∈ N and a partition (Q ,...,Q ) of Q such that q+n ≥ p. Here D = D .
1 n Qi k∈Qi k
So, each monomial in A and/or derivatives of A has at least order p. We are assuming
Q
that all these terms are in C .
1
proof: We prove Lemma 2.3 by induction. We write d(p) = d(p)(A;1).
First step: Differentiating once (1.9) gives
∞
D d(p) = (−1)j−1 T (Aj−1D A) d(p) (2.13)
1 1 1
j=p
X
Since
∞
(−1)j−1Aj−1D A = (R+ξ )D A
1 p 1
j=p
X
= R[D +ξ D A](1+A) (2.14)
1 p 1
we have from (2.12)
D d(p) = T (R A ) d(p) (2.15)
1 1 {1}
which establishes (2.11) for n = 1.
Induction step: We now assume (2.11) valid for n = k. By differentiating (2.11) with
respect to (k+1)-th variable and using (2.15) with {1} replaced by {k+1} it follows that
k+1
D d(p)
j
(cid:18)j=1 (cid:19)
Y
= D T (∧sR·A ∧···∧A )+T (∧sR·A ∧···∧A )T (RA ) d(p)
k+1 s P1 Ps s P1 Ps 1 {k+1}
PX∈Pk(cid:8) (cid:9)
(2.16)
We have
s
D T (∧sR·A ∧···∧A ) = T (∧sR·A ∧···∧D A ∧···∧A )
k+1 s P1 Ps s P1 k+1 Pi Ps
i=1
X
−T (∧sR·d∧s (RD A)·A ∧···∧A ) (2.17)
s k+1 P1 Ps
The second term in the right hand side of (2.17) can be written as
T (∧sR·d∧s(ξ D A)·A ∧···∧A )−T (∧sR·d∧(RA )·A ∧···∧A ) (2.18)
s p k+1 P1 Ps s {k+1} P1 Ps
8
Now, it follows from (2.6) that
A ∧···∧D A ∧···∧A +d∧s (ξ D A)·A ∧···∧A
P1 k+1 Pi Ps p k+1 P1 Ps
i
X
= A ∧···∧(D +ξ D A)A ∧···∧A )
P1 k+1 p k+1 Pi Ps
i
X
= A ∧···∧A ∧···∧A (2.19)
P1 Pi∪{k+1} Ps
i
X
¿From (2.16) - (2.19) we have
k+1
Dj d(p) = (WP +YP)d(p) (2.20)
(cid:18)jY=1 (cid:19) PX∈Pk
where
WP = Ts(∧sR·AP1 ∧···∧APs)T1(RA{k+1})−Ts(∧sR·d∧s (RA{k+1})·AP1 ∧···∧APs)
and
s
YP = Ts(∧sR·AP1 ∧···∧APi∪{k+1} ∧···∧APs) (2.21)
i=1
X
We now set in Lemma 2.2 A = RA for j = 1,...,k and A = RA . It
j Pj k+1 {k+1}
follows from Lemma 2.1, Lemma 2.2 and Remark 2.4 that
WP = Ts+1(∧s+1R·AP1 ∧···∧APs ∧A{k+1}) (2.22)
¿From (2.21) and (2.22) we can write (2.20) as
T (∧sR·A ∧···∧A )d(p) (2.23)
s P1 Ps
P∈XPk+1
which proves that (2.11) is also valid for n = k+1, completes our induction argument and
proves Lemma 2.3.
9
3. PROOF OF THEOREMS 1.1 AND 1.4
Let D = ... = D = d and A = λK in Lemma 2.4. Since we have for any
1 n dλ
Q ⊂ {1,...,n}
|Q|
d
[K ] = +ξ (λK)K (1+λK)
Q λ=0 dλ p
" #
(cid:18) (cid:19) λ=0
|Q|
d
= +ξ (λ) (1+λ) K|Q| (3.1)
p
dλ
" #
(cid:18) (cid:19) λ=0
(1.12) follows from (2.11).
Let us assume that R(K;λ)d(p)(K;λ) is an analytic (entire) operator valued function
of λ. We have from (1.2) that
n
d
D(p)(K) = R(K;λ)d(p)(K;λ)
n dλ
(cid:20)(cid:18) (cid:19) (cid:21)λ=0
(cid:16) (cid:17)
n+1
1 n+1
= (−1)m−1m! Km−1 C(p) T (K|P1| ∧···∧K|Ps|)
n+1 mX=1(cid:18) m (cid:19) P∈PXn+1−m Yi |Pi|! s
(3.2)
from which we get (1.13).
We shall now establish the analytic properties of d(p) and D(p). We begin with the
determinant.
Let P be the collection of partitions P = (P ,...P ) of {1,...,n}. We define
n 1 s
T(p) = t(p) ...t(p) (3.3)
n |P1| |Ps|
PX∈Pn
where
0 if r < p
t(p) = (3.4)
r cr p−2r ! kKkr otherwise
( p−1 r
(cid:16) (cid:17)
It follows from (1.16) - (1.18) that
|d(p)(K)| ≤ T(p) (3.5)
n n
Let k = #{P : |P | = j,i = 1,...,s}. Then (3.3) can be written as
j i i
(p) k1 (p) kn
1 t t
T(p) = n! 1 ... n (3.6)
n k !···k ! 1! n!
1 n ! !
k1X,...,kn
10