# Perturbations of Regularized Determinants of  Operators in a Banach Space

Authors: Michael Gil
Journal: Journal of Mathematics
 (2013)

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## Abstract

LetX be a separable Banach space with the approximation property. For an integer p≥1, let Γp be a quasinormed ideal of compact operators in X with a quasinorm NΓp, such that ∑k=1∞|λk(A)|p≤apNΓpp(A)(A∈Γp), where λk(A) are the eigenvalues of A and ap is a constant independent of A. We suggest upper and lower bounds for the regularized determinants of operators from Γp as well as bounds for the difference between determinants of two operators. Applications to the p-summing operators, Hille-Tamarkin integral operators, Hille-Tamarkin matrices, Schatten-von Neumann operators, and Lorentz operator ideals are discussed.


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## Body

## 1. Statement of the Main Result

LetX be a separable Banach space with the approximation property and the unit operator I. Let Ep(z) be the Weierstrass primary factor:
(1)E1(z)=(1-z),Ep(z)=(1-z)exp[∑m=1p-1‍zmm](p=2,3,…;z∈ℂ).
For a Riesz operator A whose eigenvalues counted with their algebraic multiplicities are denoted by λk(A)  (k=1,2,…), introduce the p-regularized determinant
(2)detp(I-A)=∏k=1∞Ep(λk(A)),
provided
(3)∑k=1∞|λk(A)|p<∞.
The classical theory of regularized determinants for Schatten-von Neumann operators has a long history, which is presented, in particular, in [1, 2]. König [3] developed the theory of regularized determinants for absolutely p-summing operators (2<p<∞) in a Banach space. In [2, 4], following the classical pattern, regularized determinants are defined for operators of the form I+A, in a Banach space where not necessarily A itself but at least some power Am admits a trace. The idea is to replace in all formulas the undefined traces by zero.LetSNp(p=1,2,…) be the von Neumann-Schatten ideal of compact operators A in a separable Hilbert space H with the finite norm Np(A)=[Trace(AA*)p/2]1/p, where A* is adjoint to A. The following inequalities are well-known:
(4)|detp(I-A)|≤exp[dpNpp(A)],|detp(I-A)-detp(I-B)|≤Np(A-B)exp[dp(1+[Np(A)+Np(B)])p]
with an unknown constant dk, see the books [1, page 1106] and [2, page 194]. In [5, 6] these inequalities were slightly improved. In [7] it was proved that one can take dp=γp, where
(5)γp:=p-1p(p≠1;p≠3),γ1=γ3=1.In this paper we investigate a quasinormed idealΓp of compact operators in X with a quasinorm NΓp(·). That is, NΓp(·) satisfies all the usual properties of a norm, with the exception of the triangular inequality, which is replaced by
(6)NΓp(A+B)≤b(Γp)(NΓp(A)+NΓp(B))
with a constant b(Γp) independent of A,B. Moreover, for an integer 1≤p<∞ and any A∈Γp, the inequality
(7)∑k=1∞|λk(A)|p≤apNΓpp(A)
holds, where ap is a constant independent of A (but dependent on Γp). The aim of this paper is to generalize inequalities (4) to operators from Γp. In addition, a lower bound for detp(I-A) is established.Now we are in a position to formulate our main result.Theorem 1.
LetA,B∈Γp for an integer p≥1. Then
(8)|detp(I-A)|≤eapγpNΓpp(A),(9)|detp(I-A)-detp(I-B)|≤NΓp(A-B)×exp[γpapb(Γp)(1+12[NΓp(A-B)+NΓp(A+B)])p].This theorem is proved in the next section.Note that ifNΓp is a norm, then b(Γp)=1; if Γp=SNp, then ap=1.


## 2. Proof of Theorem1

LetY be an quasinormed space, that is, it is a linear space with a quasinorm N(·)=NY(·). Namely,
(10)N(C+C~)≤c(N(C)+N(C~))(C,C~∈Y)
with a constant c=cY.Lemma 2.
For allC,C~∈Y, let f(C+λC~) be a scalar-valued entire function of λ∈ℂ and there be a monotone nondecreasing function G:[0,∞)→[0,∞), such that
(11)|f(C)|≤G(N(C))
for all C∈Y. Then
(12)|f(C)-f(C~)|≤N(C-C~)G(c(1+12N(C-C~)+12N(C+C~))).Proof.
Put(13)g1(λ)=f(12(C+C~)+λ(C-C~)).
Theng1(λ) is an entire function and
(14)f(C)-f(C~)=g1(12)-g1(-12).
Thanks to the Cauchy integral,
(15)g1(12)-g1(-12)=12πi∫|z|=1/2+r‍g1(z)dz(z-1/2)(z+1/2)(r>0).
Hence,
(16)|g1(12)-g1(-12)|≤(12+r)sup|z|=1/2+r|g1(z)||z2-(1/4)|≤1rsup|z|=1/2+r|g1(z)|.
In addition, by (11),
(17)|g1(z)|=|f(12(C+C~)+z(C-C~))|≤G(N(12(C+C~)+z(C-C~)))≤G(12cN(C+C~)+(12+r)cN(C-C~))(|z|=12+r).
Therefore according to (15),
(18)|f(C)-f(C~)|=|g1(12)-g1(-12)|≤1rG(12cN(C+C~)+(12+r)cN(C-C~)).
Taking r=1/N(C-C~), we get the required result.We need also the following result, proved in [7, Lemma 2.3].Lemma 3.
For any integerp≥1 and all z∈ℂ, one has |Ep(z)|≤exp[γp|z|p].Proof of Theorem1.
By the previous lemma(19)|detp(I-A)|=∏k=1∞|Ep(λk(A))|≤∏k=1∞eγp|λk(A)|p=exp[γp∑k=1∞|λk(A)|p].
Now (8) follows from the latter inequality and (7).
Moreover, (8) and Lemma 2 imply (9).


## 3. Lower Bounds

Let1∉σ(A) and L be  a Jordan curve connecting0  and1, lying in the disc{z∈ℂ:|z|≤1}  and such that(20)ϕA≔infs∈L;k=1,2,…|1-sλk(A)|>0.
Let l=|L| be the length of L. For example, if A does not have the eigenvalues on [1,∞), then one can take L=[0,1]. In this case l=1 and
(21)ϕA=infs∈[0,1];k=1,2,…|1-sλk(A)|.
If the spectral radius rs(A) of A is less than one, then l=1, ϕA≥1-rs(A).Theorem 4.
LetA∈Γp(p=1,2,…), 1∉σ(A), and condition (20) hold. Then
(22)|detp(I-A)|≥e-(lapNΓpp(A)/ϕA).Proof .
We have(23)detp(I-zA)=∏j=1∞Ep(zλj)(λj=λj(A)).
Clearly,
(24)ddzdetp(I-zA)=∑k=1∞dEp(zλk)dz∏j=1,j≠k∞Ep(zλj),dEp(zλj)dz=[-λj+(1-zλj)∑m=0p-2‍zmλjm+1]exp[∑m=1p‍zmλjmm].
But
(25)-λj+(1-zλj)∑m=0p-2‍zmλjm+1=-zp-1λjp,
since
(26)∑m=0p-2‍zmλjm=1-(zλj)p-11-zλj.
So
(27)dEp(zλj)dz=-zp-1λjpexp[∑m=1p‍zmλjmm]=-zp-1λjp1-zλjEp(zλj).
Hence,
(28)ddzdetp(I-zA)=h(z)detp(I-zA),
where
(29)h(z)≔-zp-1∑k=1∞λkp(A)1-zλk(A).
Consequently,
(30)detp(I-A)=exp[∫L‍h(s)ds].
But |s|≤1 for any s∈L, and thus by (7)
(31)|∫L‍h(s)ds|≤∑k=1∞|λk(A)|p∫L‍|s|p-1|ds||1-sλk(A)|≤lϕA∑k=1∞|λk(A)|p≤lapϕANΓpp(A).
Therefore,
(32)|detp(I-A)|=|exp[∫L‍h(s)ds]|≥exp[-|∫L‍h(s)ds|]≥exp[-NΓpplapϕA-1(A)],
as claimed.Since(33)|detp(I-B)|≥|detp(I-A)|-|detp(I-A)-detp(I-B)|,
Theorems 1 and 4 imply the following result.Corollary 5.
LetA,B∈Γp for an integer p≥1, 1∉σ(A), and condition (20) hold. If, in addition,
(34)exp[-aplNΓpp(A)ϕA]>NΓp(A-B)×exp[apγpb(Γp)(1+12(NΓp(A+B)+NΓp(A-B)))p],
then I-B is invertible.


## 4. Applications

Suppose1≤p<∞ and that A a linear operator in X. A is said to be p-summing, if there is a constant ν such that regardless of a natural number m and regardless of the choice x1,…,xm∈X we have
(35)[∑k=1m‍∥Axk∥p]1/p≤νsup{[∑k=1m‍|(x*,xk)|p]1/p:x*∈X*,∥x*∥=1},
cf. [8]. The least ν for which this inequality holds is denoted by πp(A). The set of p-summing operators in X with the finite norm πp is an ideal, cf. [9], which is denoted by Πp. By the well-known Theorem  3.7.2 in [9, page 159],
(36)∑k=1∞|λk(A)|p≤πpp(A)(A∈Πp;2≤p<∞)
(see also Theorem  17.4.3 in [10, page 298]). Since πp(A) is a norm, Theorems 1 and  4 imply  the following.Corollary 6.
LetA,B∈Πp for some integer p≥2. Then |detp(I-A)|≤exp[γpπpp(A)] and
(37)|detp(I-A)-detp(I-B)|≤πp(A-B)exp[γp(1+12[πp(A-B)+πp(A+B)])p].
If, in addition, (20) holds, then
(38)|detp(I-A)|≥e-(lπpp(A)/ϕA).Furthermore, letLμp(Ω)(Ω⊂ℝn;1<p<∞) be the space of scalar functions f defined on Ω with a finite positive measure μ and the norm
(39)∥f∥=[∫Ω‍|f(x)|pdμ]1/p.
Let K:Lμp(Ω)→Lμp(Ω) be the integral operator
(40)(Kf)(t)=∫Ω‍k(t,s)f(s)dμ
whose kernel k defined on Ω×Ω satisfies the condition
(41)k^p(K)≔[∫Ω‍(∫Ω‍|k(t,s)|p′dμ(s))p/p′dμ(t)]1/p<∞,
where 1/p+1/p′=1. Then K is called a (p,p′)-Hille-Tamarkin operator. As it is well known [8, page 43], any (p,p′)-Hille-Tamarkin operator K is a p-summing operator and
(42)πp(K)≤k^p(K).
Since k^p(·) is a norm, by Theorems 1 and 4 we get.Corollary 7.
LetK and K~ be (p,p′)-Hille-Tamarkin operators in Lμp(Ω) for an integer p≥2. Then |detp(I-K)|≤exp[γpk^pp(K)] and
(43)|detp(I-K)-detp(I-K~)|≤k^p(K~-K)exp[γp(1+12[k^p(K~-K)+k^p(K~+K)])p].
If, in addition, condition (20) holds for A=K, then
(44)|detp(I-K)|≥e-(lk^pp(K)/ϕK).Now let us consider a linear operatorT in lp  (1<p<∞) generated by an infinite matrix (tjk)j,k=1∞, satisfying
(45)τ^p(T)≔[∑j=1∞(∑k=1∞|tjk|p′)p/p′]1/p<∞.
Then T is called a (p,p′)-Hille-Tamarkin matrix. As it is well known [8, page 43], any (p,p′)-Hille-Tamarkin matrix T is a p-summing operator with
(46)πp(T)≤τ^p(T),
cf. [9, Sections 5.3.2 and 5.3.3, page 230].Sinceτ^p(·) is a norm, Theorems 1 and 4 imply the following.Corollary 8.
LetT and T~ be (p,p′)-Hille-Tamarkin matrices for an integer p≥2. Then |detp(I-T)|≤exp[γpτ^pp(T)] and
(47)|detp(I-T)-detp(I-T~)|≤τ^p(T~-T)×exp[γp(1+12[τ^p(T~-T)+τ^p(T~+T)])p].
If, in addition, condition (20) holds for A=T, then
(48)|detp(I-T)|≥e-(lτ^pp(T)/ϕT).Now letX=H be a separable Hilbert space and Lq,r  (q>1,0<r<q) the Lorentz ideal of compact operators T with the finite quasinorm
(49)Nq,r(T)=[∑k=1∞k(q/r)-1skq(T)]1/q,
where sk(T) are the singular numbers of T taken with their multiplicities. So
(50)Nq,r(T+T~)≤cq,r(Nq,r(T)+Nq,r(T~))(cq,r=const;T,T~∈Lq,r).
For the details, see [11, Section 1.1]. By [11, Lemma 1.4],
(51)∑k=1∞k(q/r)-1|λk(T)|q≤cq,rNq,rq(T).
For an integer p≥1, let q>p and r=qp/(p+q). Then simple calculations show that (q/r)-1=q/p. By the Hölder inequality, for d=q/p, we obtain
(52)∑k=1∞|λk(T)|p≤τ(d)(∑k=1∞kd|λk(T)|pd)1/d
with
(53)τ(d)=(∑k=1∞k-d′)1/d′(1d′+1d=1).
So we have
(54)∑k=1∞|λk(T)|p≤τ(q/p)(∑k=1∞k(q/r)-1|λk(T)|q)p/q.
Thus (51) implies the following result.Lemma 9.
For an integerp≥1 and a q>p,  let T∈Lq,r with r=qp/(p+q). Then
(55)∑k=1∞|λk(T)|p≤cq,rτ(qp)Nq,rp(T).Now we can directly apply Theorems1 and 4.



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