1. Statement of the Main Result
Let X 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.
Let SNp (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 ideal Γp 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.
Let A,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 if NΓp is a norm, then b(Γp)=1; if Γp=SNp, then ap=1.
2. Proof of Theorem 1
Let Y 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 all C,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~)).
Then g1(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 integer p≥1 and all z∈ℂ, one has |Ep(z)|≤exp[γp|z|p].
Proof of Theorem 1.
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
Let 1∉σ(A) and L be a Jordan curve connecting 0 and 1, 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.
Let A∈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-2zmλjm+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.
Let A,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
Suppose 1≤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.
Let A,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, let Lμp(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.
Let K 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 operator T 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].
Since τ^p(·) is a norm, Theorems 1 and 4 imply the following.
Corollary 8.
Let T 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 let X=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 integer p≥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 Theorems 1 and 4.