The product formula for regularized Fredholm determinants
Thomas Britz, Alan Carey, Fritz Gesztesy, Roger Nichols, Fedor Sukochev, Dmitriy Zanin
aa r X i v : . [ m a t h . SP ] A ug THE PRODUCT FORMULA FOR REGULARIZEDFREDHOLM DETERMINANTS
THOMAS BRITZ, ALAN CAREY, FRITZ GESZTESY, ROGER NICHOLS,FEDOR SUKOCHEV, AND DMITRIY ZANIN
Abstract.
For trace class operators
A, B ∈ B ( H ) ( H a complex, separableHilbert space), the product formula for Fredholm determinants holds in thefamiliar formdet H (( I H − A )( I H − B )) = det H ( I H − A )det H ( I H − B ) . When trace class operators are replaced by Hilbert–Schmidt operators
A, B ∈B ( H ) and the Fredholm determinant det H ( I H − A ), A ∈ B ( H ), by the 2ndregularized Fredholm determinant det H , ( I H − A ) = det H (( I H − A ) exp( A )), A ∈ B ( H ), the product formula must be replaced bydet H , (( I H − A )( I H − B )) = det H , ( I H − A )det H , ( I H − B ) × exp( − tr H ( AB )) . The product formula for the case of higher regularized Fredholm determinantsdet H ,k ( I H − A ), A ∈ B k ( H ), k ∈ N , k >
2, does not seem to be easily accessibleand hence this note aims at filling this gap in the literature. Introduction
The purpose of this note is to prove a product formula for regularized (modified)Fredholm determinants extending the well-known Hilbert–Schmidt case.To set the stage, we recall that if A ∈ B ( H ) is a trace class operator on thecomplex, separable Hilbert space H , that is, the sequence of (necessarily nonnega-tive) eigenvalues λ j (cid:0) ( A ∗ A ) / (cid:1) , j ∈ N = N ∪ { } , of | A | = ( A ∗ A ) / (the singularvalues of A ), ordered in nonincreasing magnitude and counted according to theirmultiplicity, lies in ℓ ( N ), the Fredholm determinant det H ( I H − A ) associated with I H − A , A ∈ B ( H ), is given by the absolutely convergent infinite productdet H ( I H − A ) = Y j ∈ J [1 − λ j ( A )] , (1.1)where λ j ( A ), j ∈ J (with J ⊆ N an approximate index set) are the (generally,complex) eigenvalues of A ordered again with respect to nonincreasing absolutevalue and now counted according to their algebraic multiplicity.A celebrated property of det H ( I H − · ) that (like the analog of (1.1)) is sharedwith the case where H is finite-dimensional, is the product formuladet H (( I H − A )( I H − B )) = det H ( I H − A )det H ( I H − B ) , A, B ∈ B ( H ) (1.2) Date : August 4, 2020.2020
Mathematics Subject Classification.
Primary: 47B10; Secondary: 47B02.
Key words and phrases.
Trace ideals, regularized Fredholm determinants, determinant productformula.A.L.C., G.L. and F.S. gratefully acknowledge the support of the Australian Research Council. (see, e.g., [4, pp. 162–163], [7, Theorem XIII.105 ( a )], [8, Theorem 3.8], [9, Theo-rem 3.5 ( a )], [10, Theorem 3.4.10 ( f )], [11, p. 44]).When extending these considerations to operators A ∈ B p ( H ), with B p ( H ), p ∈ [1 , ∞ ), the ℓ p ( N )-based trace ideals (i.e., the eigenvalues λ j (cid:0) ( A ∗ A ) / (cid:1) , j ∈ N ,of ( A ∗ A ) / now lie in ℓ p ( N ), see, e.g., [4, Sect. III.7]), the k th regularized Fredholmdeterminant det H ,k ( I H − A ), k ∈ N , associated with I H − A , A ∈ B k ( H ), is givenby det H ,k ( I H − A ) = Y j ∈ J (cid:18) [1 − λ j ( A )] exp (cid:18) k − X ℓ =1 ℓ − λ j ( A ) ℓ (cid:19)(cid:19) = det H (cid:18) ( I H − A ) exp (cid:18) k − X ℓ =1 ℓ − A ℓ (cid:19)(cid:19) (1.3)(see, e.g., [2, pp. 1106–1116], [4, pp. 166–169], [8], [9, pp. 75–76], [10, pp. 187–191],[11, p. 44]).We note that det H ,k ( I H − · ) is continuous on B ℓ ( H ) for 1 ℓ k , anddet H ,k ( I H − AB ) = det H ,k ( I H − BA ) , A, B ∈ B ( H ) , AB, BA ∈ B k ( H ) (1.4)(this extends to the case where A maps between different Hilbert spaces H and H and B from H to H , etc.).The analog of the simple product formula (1.2) no longer holds for k > k = 2 that (1.2) must be replacedbydet H , (( I H − A )( I H − B )) = det H , ( I H − A )det H , ( I H − B ) exp( − tr H ( AB )) ,A, B ∈ B ( H ) (1.5)(see, e.g., [4, p. 169], [9, p. 76], [10, p. 190], [11, p. 44]). Recently, some of us neededthe extension of (1.5) to general k ∈ N , k >
3, in [1], but were not able to find it inthe literature; hence, this note aims at closing this gap.More precisely, we were interested in a product formula for det H ,k (( I H − A )( I H − B )) for A, B ∈ B k ( H ) in terms of det H ,k ( I H − A ) and det H ,k ( I H − B ), k ∈ N , k > A is a finiterank operator, denoted by F , and B ∈ B k ( H ) was considered in [5, Lemma 1.5.10](see also, [6, Proposition 4.8 ( ii )]), and the resultdet H ,k (( I H − F )( I H − B )) = det H ( I H − F )det H ,k ( I H − B ) exp (tr H ( p n ( F, B ))) , (1.6)with p n ( · , · ) a polynomial in two variables and of finite rank, was derived. Anextension of this formula to three factors, that is,det H ,k (( I H − A )( I H − F )( I H − B )) = det H ( I H − F )det H ,k (( I H − A )( I H − B )) × exp (tr H ( p n ( A, F, B ))) , (1.7)with p n ( · , · , · ) a polynomial in three variables and of finite rank, was derived in[3, Lemma C.1].The result we have in mind is somewhat different from (1.6) in that we areinterested in a quantitative version of the following fact: Theorem 1.1.
Let k ∈ N , and suppose A, B ∈ B k ( H ) . Then det H ,k (( I H − A )( I H − B )) = det H ,k ( I H − A )det H ,k ( I H − B ) exp(tr H ( X k ( A, B ))) , (1.8) HE PRODUCT FORMULA FOR REGULARIZED FREDHOLM DETERMINANTS 3 where X k ( · , · ) ∈ B ( H ) is of the form X ( A, B ) = 0 ,X k ( A, B ) = k − X j ,...,j k − =0 c j ,...,j k − C j · · · C j k − k − , k > , (1.9) with c j ,...,j k − ∈ Q ,C ℓ = A or B, ℓ k − ,k k − X ℓ =1 j ℓ k − , k > . (1.10)Explicitly, one obtains: X ( A, B ) = 0 ,X ( A, B ) = − AB,X ( A, B ) = 2 − (cid:2) ( AB ) − AB ( A + B ) − ( A + B ) AB (cid:3) , (1.11) X ( A, B ) = 2 − ( AB ) − − (cid:2) AB ( A + B ) + ( A + B ) AB + ( A + B ) AB ( A + B ) (cid:3) + 3 − (cid:2) ( AB ) ( A + B ) + ( A + B )( AB ) + AB ( A + B ) AB (cid:3) − − ( AB ) , etc.When taking traces (what is actually needed in (1.8)), this simplifies totr H ( X ( A, B )) = 0 , tr H ( X ( A, B )) = − tr H ( AB ) , tr H ( X ( A, B )) = − tr H (cid:0) ABA + BAB − − ( AB ) (cid:1) , tr H ( X ( A, B )) = − tr H (cid:0) A B + A B + AB + 2 − ( AB ) − ( AB ) A − B ( AB ) + 3 − ( AB ) (cid:1) , etc. (1.12)We present the proof of a quantitative version of Theorem 1.1 in two parts. In thenext section we prove an algebraic result, Lemma 2.4, that is the key to the analyticpart of the argument appearing in the final section on regularized determinants.2. The Commutator Subspace in the Algebra ofNoncommutative Polynomials
To prove a quantitative version of Theorem 1.1 and hence derive a formula for X k ( A, B ), we first need to recall some facts on the commutator subspace of analgebra of noncommutative polynomials.Let Pol be the free polynomial algebra in 2 (noncommuting) variables, A and B .Let W be the set of noncommutative monomials (words in the alphabet { A, B } ).(We recall that the set W is a semigroup with respect to concatenation, 1 is theneutral element of this semigroup, that is, 1 is an empty word in this alphabet.) T. BRITZ ET AL.
Every x ∈ Pol can be written as a sum x = X w ∈ W b x ( w ) w. (2.1)Here the coefficients b x ( w ) vanish for all but finitely many w ∈ W .Let [Pol , Pol ] be the commutator subspace of Pol , that is, the linear span ofcommutators [ x , x ], x , x ∈ Pol . Lemma 2.1.
One has x ∈ [Pol , Pol ] provided that X w ′ ∼ w b x ( w ′ ) = 0 , w ∈ W. (2.2) Here, w ′ ∼ w stands for the fact that the word w ′ is obtained from w by a cyclicpermutation.Proof. Let L ( w ) be the length of the word w . Then x = X w ∈ W b x ( w ) w = b x (1) + X w =1 L ( w ) − X w ′ ∼ w b x ( w ′ ) w ′ . (2.3)Obviously, ( w ′ − w ) ∈ [Pol , Pol ] whenever w ′ ∼ w and thus, x ∈ (cid:18)b x (1) + X w =1 L ( w ) − X w ′ ∼ w b x ( w ′ ) w + [Pol , Pol ] (cid:19) . (2.4)By hypothesis, b x (1) = 0 and X w ′ ∼ w b x ( w ′ ) = 0 , = w ∈ W, (2.5)completing the proof. (cid:3) Next, we need some notation. Let k , k ∈ N = N ∪ { } , and set z k ,k = , k = k = 0 ,k − A k , k ∈ N , k = 0 ,k − B k , k = 0 , k ∈ N , P k + k j =1 j − P π ∈ S j , | π | =3 | π | + | π | = k | π | + | π | = k ( − | π | z π , k , k ∈ N . (2.6)Here, S j is the set of all partitions of the set { , · · · , j } , 1 j k + k . (Thesymbol | · | abbreviating the cardinality of a subset of Z .) The condition | π | = 3means that π breaks the set { , · · · , j } into exactly 3 pieces denoted by π , π , and π (some of them can be empty). The element z π denotes the product z π = j Y m =1 z m,π , z m,π = A, m ∈ π ,B, m ∈ π ,AB, m ∈ π . (2.7)Finally, let W k ,k be the collection of all words with k letters A and k letters B .Using this notation we now establish a combinatorial fact. HE PRODUCT FORMULA FOR REGULARIZED FREDHOLM DETERMINANTS 5
Lemma 2.2.
Let k , k ∈ N . Then z k ,k = X w ∈ W k ,k (cid:18) n ( w ) X ℓ =0 ( − ℓ k + k − ℓ (cid:18) n ( w ) ℓ (cid:19)(cid:19) w, (2.8) where n ( w ) = | S ( w ) | , S ( w ) = { ℓ L ( w ) − | w ℓ = A, w ℓ +1 = B } . (2.9) Proof.
For each j ∈ { , . . . , k + k } , letΠ j = { π ∈ S j | | π | = 3 , | π | + | π | = k , | π | + | π | = k } , (2.10)Π j,w = { π ∈ Π j | z π = w } , w ∈ W k ,k . (2.11)One observes that | π | n ( w ) min { k , k } and that j = | π | + | π | + | π | = k + k − | π | . (2.12)For any partition π ∈ Π j,w , let I ⊆ S ( w ) indicate which subwords AB in w arisefrom elements in π . Then | I | = | π | = k + k − j . Therefore, each partition in π ∈ Π j,w is determined by a unique choice of I and each such choice of I determinesthe choice of π uniquely. This implies that | Π j,w | = (cid:18) n ( w ) k + k − j (cid:19) . (2.13)Thus, z k ,k = X w ∈ W k ,k k + k X j =1 j − X π ∈ Π j,w ( − | π | w = X w ∈ W k ,k k + k X j =1 j − X π ∈ Π j,w ( − | π | w = X w ∈ W k ,k k + k X j =1 ( − k + k − j j − | Π j,w | w = X w ∈ W k ,k k + k X j =1 ( − k + k − j j − (cid:18) n ( w ) k + k − j (cid:19) w. (2.14)Taking into account that (cid:18) n ( w ) k + k − j (cid:19) = 0 , k + k − j / ∈ { , · · · , n ( w ) } , (2.15)it follows that z k ,k = X w ∈ W k ,k k + k X j = k + k − n ( w ) ( − k + k − j j − (cid:18) n ( w ) k + k − j (cid:19) w = X w ∈ W k ,k (cid:18) n ( w ) X ℓ =0 ( − ℓ k + k − ℓ (cid:18) n ( w ) ℓ (cid:19)(cid:19) w. (2.16) (cid:3) T. BRITZ ET AL.
We can now prove the main fact about the commutator subspace of Pol neededlater on. Lemma 2.3.
For every k , k ∈ N , z k ,k ∈ [Pol , Pol ] .Proof. For any word w = w w · · · w k + k ∈ W k ,k we define the cyclic shift σ ( w ) = w · · · w k + k w . Suppose that w starts with the subword w w = AB . Consideringtwo cyclic shifts of w , w = AB · · · , σ ( w ) = B · · · A, σ ( σ ( w )) = · · · AB, (2.17)one sees that n ( σ ( w )) = n ( w ) − n ( σ ( σ ( w ))) = n ( w ). Hence, the sequence (cid:8) n (cid:0) σ m w (cid:1)(cid:9) m Next, we introduce some further notation. Let k ∈ N and set x = 0 ,x k = k − X j =1 j − X A⊆{ , ··· ,j } j + |A| > k ( − |A| y A , k > , (2.23) y = 0 ,y k = k − X j =1 j − X A⊆{ , ··· ,j } j + |A| k − ( − |A| y A , k > , (2.24) y A = j Y m =1 y m, A , y m, A = ( A + B, m / ∈ A ,AB, m ∈ A . (2.25)In particular, k − X j =1 j − ( A + B − AB ) j = x k + y k , (2.26)and one notes that the length of the word y A subject to A ⊆ { , . . . , j } , equals L ( y A ) = (cid:12)(cid:12) A c (cid:12)(cid:12) + 2 |A| = j + |A| , j k − , k > A c = { , . . . , j }\A the complement of A in { , . . . , j } ).Using this notation we can now state the following fact: Lemma 2.4. Let k ∈ N , k > , then y k ∈ (cid:18) k − X j =1 j ( A j + B j ) + [Pol , Pol ] (cid:19) . (2.28) Proof. Employing y k = X k ,k > k + k k − z k ,k , (2.29)Lemma 2.3 yields z k ,k ∈ [Pol , Pol ] , k , k ∈ N . (2.30)Since by (2.6), z , = 0 , z k , = k − A k , k ∈ N , z ,k = k − B k , k ∈ N , (2.31)combining (2.29)–(2.31) completes the proof. (cid:3) The Product Formula for k th Modified Fredholm Determinants After these preparations we are ready to return to the product formula for reg-ularized determinants and specialize the preceding algebraic considerations to thecontext of Theorem 1.1.First we recall that by (2.23) and (2.27), x k = k − X j =1 j − X A⊆{ , ··· ,j } j + |A| > k ( − |A| y A := X k ( A, B ) ∈ B ( H ) , k > , (3.1) T. BRITZ ET AL. since for 1 j k − L ( y A ) = j + |A| > k , and hence one obtains the inequality k x k k B ( H ) c k max k ,k 2. We also set (cf. (2.23) X ( A, B ) = 0. Theorem 3.1. Let k ∈ N and assume that A, B ∈ B k ( H ) . Then det H ,k (( I H − A )( I H − B )) = det H ,k ( I H − A )det H ,k ( I H − B ) exp(tr H ( X k ( A, B ))) . (3.3) Proof. First, we suppose that A, B ∈ B ( H ). Then it is well-known thatdet H ,k ( I H − A )det H ,k ( I H − B ) = det H ,k (( I H − A )( I H − B )) , (3.4)consistent with X ( A, B ) = 0. Without loss of generality we may assume that k ∈ N , k > 2, in the following. Employingdet H ,k ( I H − T ) = det H ( I H − T ) exp (cid:18) tr H (cid:18) k − X j =1 j − T j (cid:19)(cid:19) , T ∈ B k ( H ) , (3.5)one infers thatdet H ,k (( I H − A )( I H − B )) = det H ,k ( I H − ( A + B − AB ))= det H ( I H − ( A + B − AB )) exp (cid:18) tr H (cid:18) k − X j =1 j − ( A + B − AB ) j (cid:19)(cid:19) = det H ( I H − A )det H ( I H − B ) exp (cid:18) tr H (cid:18) k − X j =1 j − ( A + B − AB ) j (cid:19)(cid:19) = det H ,k ( I H − A )det H ,k ( I H − B ) × exp (cid:18) tr H (cid:18) k − X j =1 j − (cid:2) ( A + B − AB ) j − A j − B j (cid:3)(cid:19)(cid:19) . (3.6)By (2.26) one concludes thattr H (cid:18) k − X j =1 j − (cid:2) ( A + B − AB ) j − A j − B j (cid:3)(cid:19) = tr H ( x k )+tr H (cid:18) y k − k − X j =1 j − (cid:0) A j + B j (cid:1)(cid:19) . (3.7)By Lemma 2.4, y k − k − X j =1 j − (cid:0) A j + B j (cid:1) (3.8)is a sum of commutators of polynomial expressions in A and B . Hence, (cid:18) y k − k − X j =1 j − (cid:0) A j + B j (cid:1)(cid:19) ⊂ [ B ( H ) , B ( H )] , (3.9)and thus, tr H (cid:18) y k − k − X j =1 j − (cid:0) A j + B j (cid:1)(cid:19) = 0 , (3.10)proving assertion (3.3) for A, B ∈ B ( H ). HE PRODUCT FORMULA FOR REGULARIZED FREDHOLM DETERMINANTS 9 Since both, the right and left-hand sides in (3.3) are continuous with respectto the norm in B k ( H ), and B ( H ) is dense in B k ( H ), (3.3) holds for arbitrary A, B ∈ B k ( H ). (cid:3) Acknowledgments. We are indebted to Galina Levitina for very helpful remarkson a first draft of this paper and to Rupert Frank for kindly pointing out references[3] and [5] to us. References [1] A. Carey, F. Gesztesy, G. Levitina, R. Nichols, F. Sukochev, and D. Zanin, On the limitingabsorption principle for massless Dirac operators and properties of spectral shift functions ,preprint, 2020.[2] N. Dunford and J. Schwartz, Linear operators. Part II. Spectral theory. Selfadjoint operatorsin Hilbert space , Wiley & Sons, New York, 1988.[3] R. Frank, Eigenvalue bounds for Schr¨odinger operators with complex potentials. III , Trans.Amer. Math. Soc. , 219–240 (2017).[4] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Opera-tors , Translations of Mathematical Monographs, Vol. 18, Amer. Math. Soc., Providence, RI,1969.[5] M. Hansmann, On the discrete spectrum of linear operators in Hilbert spaces , Ph.D. Thesis,Technical University of Clausthal, Germany, 2005.[6] M. Hansmann, Perturbation determinants in Banach spaces – with an application to eigen-value estimates for perturbed operators , Math. Nachr. , 1606–1625 (2016).[7] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators ,Academic Press, New York, 1978.[8] B. Simon, Notes on infinite determinants of Hilbert space operators , Adv. Math. , 244–273(1977).[9] B. Simon, Trace Ideals and Their Applications , Mathematical Surveys and Monographs, Vol.120, 2nd ed., Amer. Math. Soc., Providence, RI, 2005.[10] B. Simon, Operator Theory. A Comprehensive Course in Analysis, Part 4 , American Math.Soc., Providence, RI, 2015.[11] D. R. Yafaev, Mathematical Scattering Theory. General Theory , Amer. Math. Soc., Provi-dence, RI, 1992. School of Mathematics and Statistics, UNSW, Kensington, NSW 2052, Australia E-mail address : [email protected] URL : https://research.unsw.edu.au/people/dr-thomas-britz Mathematical Sciences Institute, Australian National University, Kingsley St., Can-berra, ACT 0200, Australia and School of Mathematics and Applied Statistics, Univer-sity of Wollongong, NSW, Australia, 2522 E-mail address : [email protected] URL : http://maths.anu.edu.au/~acarey/ Department of Mathematics, Baylor University, One Bear Place E-mail address : [email protected] URL : Department of Mathematics, The University of Tennessee at Chattanooga, 415 EMCSBuilding, Dept. 6956, 615 McCallie Ave, Chattanooga, TN 37403, USA E-mail address : [email protected] URL : School of Mathematics and Statistics, UNSW, Kensington, NSW 2052, Australia E-mail address : [email protected] URL : https://research.unsw.edu.au/people/scientia-professor-fedor-sukochev School of Mathematics and Statistics, UNSW, Kensington, NSW 2052, Australia E-mail address : [email protected] URL ::