Perturbations of Regularized Determinants of Operators in a Banach Space

Abstract

Let 𝑋 be a separable Banach space with the approximation property. For an integer 𝑝≥1, let Γ_𝑝 be a quasinormed ideal of compact operators in 𝑋 with a quasinorm 𝑁_Γ𝑝, such that ∑^∞_𝑘=1|𝜆_𝑘(𝐴)|^𝑝≤𝑎_𝑝𝑁^𝑝_Γ𝑝(𝐴)(𝐴∈Γ_𝑝), where 𝜆_𝑘(𝐴) are the eigenvalues of 𝐴 and 𝑎_𝑝 is a constant independent of 𝐴. We suggest upper and lower bounds for the regularized determinants of operators from Γ_𝑝 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.

1. Statement of the Main Result

Let 𝑋 be a separable Banach space with the approximation property and the unit operator 𝐼. Let 𝐸_𝑝(𝑧) be the Weierstrass primary factor: 𝐸₁(𝑧)=(1−𝑧),𝐸_𝑝(𝑧)=(1−𝑧)exp^𝑝−1_𝑚=1𝑧^𝑚𝑚𝑝=2,3,…;𝑧∈ℂ. (1) For a Riesz operator 𝐴 whose eigenvalues counted with their algebraic multiplicities are denoted by 𝜆_𝑘(𝐴)  (𝑘=1,2,…), introduce the 𝑝-regularized determinant det_𝑝(𝐼−𝐴)=^∞_𝑘=1𝐸_𝑝𝜆_𝑘(𝐴), (2) provided ^∞_𝑘=1𝜆_𝑘(𝐴)^𝑝<∞. (3) 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 𝐴^𝑚 admits a trace. The idea is to replace in all formulas the undefined traces by zero.

Let 𝑆𝑁_𝑝(𝑝=1,2,…) be the von Neumann-Schatten ideal of compact operators 𝐴 in a separable Hilbert space 𝐻 with the finite norm 𝑁_𝑝(𝐴)=[Trace(𝐴𝐴^∗)^𝑝/2]^1/𝑝, where 𝐴^∗ is adjoint to 𝐴. The following inequalities are well-known: det_𝑝(𝐼−𝐴)≤exp𝑑_𝑝𝑁^𝑝_𝑝(𝐴),det_𝑝(𝐼−𝐴)−det_𝑝(𝐼−𝐵)≤𝑁_𝑝(𝐴−𝐵)exp𝑑_𝑝1+𝑁_𝑝(𝐴)+𝑁_𝑝(𝐵)^𝑝 (4) with an unknown constant 𝑑_𝑘, 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 𝑑_𝑝=𝛾_𝑝, where 𝛾_𝑝∶=𝑝−1𝑝𝑝≠1;𝑝≠3,𝛾₁=𝛾₃=1. (5)

In this paper we investigate a quasinormed ideal Γ_𝑝 of compact operators in 𝑋 with a quasinorm 𝑁_Γ𝑝(⋅). That is, 𝑁_Γ𝑝(⋅) satisfies all the usual properties of a norm, with the exception of the triangular inequality, which is replaced by 𝑁_Γ𝑝(𝐴+𝐵)≤𝑏Γ_𝑝𝑁_Γ𝑝(𝐴)+𝑁_Γ𝑝(𝐵) (6) with a constant 𝑏(Γ_𝑝) independent of 𝐴,𝐵. Moreover, for an integer 1≤𝑝<∞ and any 𝐴∈Γ_𝑝, the inequality ^∞_𝑘=1𝜆_𝑘(𝐴)^𝑝≤𝑎_𝑝𝑁^𝑝_Γ𝑝(𝐴) (7) holds, where 𝑎_𝑝 is a constant independent of 𝐴 (but dependent on Γ_𝑝). The aim of this paper is to generalize inequalities (4) to operators from Γ_𝑝. In addition, a lower bound for det_𝑝(𝐼−𝐴) is established.

Now we are in a position to formulate our main result.

Theorem 1. Let 𝐴,𝐵∈Γ_𝑝 for an integer 𝑝≥1. Then det_𝑝(𝐼−𝐴)≤𝑒^{𝑎𝑝𝛾𝑝𝑁𝑝Γ𝑝(𝐴)},det_𝑝(𝐼−𝐴)−det_𝑝(𝐼−𝐵)≤𝑁_Γ𝑝()𝛾𝐴−𝐵×exp_𝑝𝑎_𝑝𝑏Γ_𝑝11+2𝑁_Γ𝑝(𝐴−𝐵)+𝑁_Γ𝑝(𝐴+𝐵)^𝑝. (8)(9)

This theorem is proved in the next section.

Note that if 𝑁_Γ𝑝 is a norm, then 𝑏(Γ_𝑝)=1; if Γ_𝑝=𝑆𝑁_𝑝, then 𝑎_𝑝=1.

2. Proof of Theorem 1

Let 𝑌 be an quasinormed space, that is, it is a linear space with a quasinorm 𝑁(⋅)=𝑁_𝑌(⋅). Namely, 𝑁𝐶+𝐶≤𝑐𝑁(𝐶)+𝑁𝐶𝐶,𝐶∈𝑌 (10) with a constant 𝑐=𝑐_𝑌.

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 𝑓(𝐶)≤𝐺(𝑁(𝐶)) (11) for all 𝐶∈𝑌. Then 𝑓(𝐶)𝐶𝐶𝐺𝑐1−𝑓≤𝑁𝐶−1+2𝑁𝐶+1𝐶−2𝑁𝐶+𝐶. (12)

Proof. Put 𝑔₁(𝜆)1=𝑓2𝐶+𝐶+𝜆𝐶−𝐶. (13)
Then 𝑔₁(𝜆) is an entire function and 𝑓(𝐶)−𝑓𝐶=𝑔₁12−𝑔₁−12. (14) Thanks to the Cauchy integral, 𝑔₁12−𝑔₁−12=12𝜋𝑖_{|𝑧|=1/2+𝑟}𝑔₁(𝑧)𝑑𝑧()().𝑧−1/2)(𝑧+1/2𝑟>0 (15) Hence, 𝑔₁12−𝑔₁−12≤12+𝑟sup_{|𝑧|=1/2+𝑟}𝑔₁(𝑧)𝑧²−(≤11/4)𝑟sup_{|𝑧|=1/2+𝑟}𝑔₁(𝑧). (16) In addition, by (11), 𝑔₁(𝑧=𝑓1)2𝐶𝐶𝑁1𝐶++𝑧𝐶−≤𝐺2𝐶𝐶1𝐶++𝑧𝐶−≤𝐺2𝐶+1𝑐𝑁𝐶+2𝐶|𝑧|=1+𝑟𝑐𝑁𝐶−2+𝑟. (17) Therefore according to (15), 𝑓(𝐶)𝐶=𝑔−𝑓₁12−𝑔₁−12≤1𝑟𝐺12𝐶+1𝑐𝑁𝐶+2+𝑟𝑐𝑁𝐶−𝐶. (18) 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 |𝐸_𝑝(𝑧)|≤exp[𝛾_𝑝|𝑧|^𝑝].

Proof of Theorem 1. By the previous lemma det_𝑝(𝐼−𝐴)=^∞_𝑘=1𝐸_𝑝𝜆_𝑘(𝐴≤_∞𝑘=1)𝑒^{𝛾𝑝|𝜆𝑘(𝐴)|𝑝}=exp𝛾_𝑝^∞_𝑘=1𝜆_𝑘(𝐴)^𝑝. (19) 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 that𝜙_𝐴∶=inf_{𝑠∈𝐿;𝑘=1,2,…}1−𝑠𝜆_𝑘(𝐴)>0. (20) 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 𝜙_𝐴=inf_{𝑠∈[0,1];𝑘=1,2,…}1−𝑠𝜆_𝑘(𝐴). (21) If the spectral radius 𝑟_𝑠(𝐴) of 𝐴 is less than one, then 𝑙=1, 𝜙_𝐴≥1−𝑟_𝑠(𝐴).

Theorem 4. Let 𝐴∈Γ_𝑝(𝑝=1,2,…), 1∉𝜎(𝐴), and condition (20) hold. Then det_𝑝(𝐼−𝐴)≥𝑒^{−(𝑙𝑎𝑝𝑁𝑝Γ𝑝(𝐴)/𝜙𝐴)}. (22)

Proof . We have det_𝑝(𝐼−𝑧𝐴)=^∞_𝑗=1𝐸_𝑝𝑧𝜆_𝑗𝜆_𝑗=𝜆_𝑗(𝐴). (23) Clearly, 𝑑𝑑𝑧det_𝑝(𝐼−𝑧𝐴)=^∞_𝑘=1𝑑𝐸_𝑝𝑧𝜆_𝑘𝑑𝑧^∞_{𝑗=1,𝑗≠𝑘}𝐸_𝑝𝑧𝜆_𝑗,𝑑𝐸_𝑝𝑧𝜆_𝑗=𝑑𝑧−𝜆_𝑗+1−𝑧𝜆_𝑗^𝑝−2_𝑚=0𝑧^𝑚𝜆_𝑗^𝑚+1exp^𝑝_𝑚=1𝑧^𝑚𝜆^𝑚_𝑗𝑚. (24) But −𝜆_𝑗+1−𝑧𝜆_𝑗^𝑝−2_𝑚=0𝑧^𝑚𝜆_𝑗^𝑚+1=−𝑧^𝑝−1𝜆^𝑝_𝑗, (25) since ^𝑝−2_𝑚=0𝑧^𝑚𝜆^𝑚_𝑗=1−𝑧𝜆_𝑗^𝑝−11−𝑧𝜆_𝑗. (26) So 𝑑𝐸_𝑝𝑧𝜆_𝑗𝑑𝑧=−𝑧^𝑝−1𝜆^𝑝_𝑗exp^𝑝_𝑚=1𝑧^𝑚𝜆^𝑚_𝑗𝑚𝑧=−^𝑝−1𝜆^𝑝_𝑗1−𝑧𝜆_𝑗𝐸_𝑝𝑧𝜆_𝑗. (27) Hence, 𝑑𝑑𝑧det_𝑝(𝐼−𝑧𝐴)=ℎ(𝑧)det_𝑝(𝐼−𝑧𝐴), (28) where ℎ(𝑧)∶=−𝑧^∞𝑝−1_𝑘=1𝜆^𝑝_𝑘(𝐴)1−𝑧𝜆_𝑘(𝐴). (29) Consequently, det_𝑝(𝐼−𝐴)=exp_𝐿ℎ(𝑠)𝑑𝑠. (30) But |𝑠|≤1 for any 𝑠∈𝐿, and thus by (7) _𝐿ℎ(𝑠)𝑑𝑠≤^∞_𝑘=1𝜆_𝑘(𝐴)^𝑝_𝐿|𝑠|^𝑝−1|𝑑𝑠|1−𝑠𝜆_𝑘(𝐴≤𝑙)𝜙_𝐴^∞_𝑘=1𝜆_𝑘(𝐴)^𝑝≤𝑙𝑎_𝑝𝜙_𝐴𝑁^𝑝_Γ𝑝(𝐴). (31) Therefore, det_𝑝(𝐼−𝐴)=exp_𝐿ℎ(𝑠)−𝑑𝑠≥exp_𝐿ℎ(𝑠)𝑑𝑠≥exp−𝑁^𝑝_Γ𝑝𝑙𝑎_𝑝𝜙_𝐴⁻¹(𝐴), (32) as claimed.

Since det_𝑝(≥𝐼−𝐵)det_𝑝(𝐼−𝐴)−det_𝑝(𝐼−𝐴)−det_𝑝(𝐼−𝐵), (33) Theorems 1 and 4 imply the following result.

Corollary 5. Let 𝐴,𝐵∈Γ_𝑝 for an integer 𝑝≥1, 1∉𝜎(𝐴), and condition (20) hold. If, in addition, −𝑎exp^𝑝𝑙𝑁^𝑝_Γ𝑝(𝐴)𝜙_𝐴>𝑁_Γ𝑝()𝑎𝐴−𝐵×exp_𝑝𝛾_𝑝𝑏Γ_𝑝11+2𝑁_Γ𝑝(𝐴+𝐵)+𝑁_Γ𝑝(𝐴−𝐵)^𝑝, (34) 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,…,𝑥_𝑚∈𝑋 we have ^𝑚_𝑘=1𝐴𝑥_𝑘^𝑝1/𝑝≤𝜈sup^𝑚_𝑘=1𝑥^∗,𝑥_𝑘^𝑝1/𝑝∶𝑥^∗∈𝑋^∗,𝑥^∗=1, (35) cf. [8]. The least 𝜈 for which this inequality holds is denoted by 𝜋_𝑝(𝐴). The set of 𝑝-summing operators in 𝑋 with the finite norm 𝜋_𝑝 is an ideal, cf. [9], which is denoted by Π_𝑝. By the well-known Theorem  3.7.2 in [9, page 159], ^∞_𝑘=1𝜆_𝑘(𝐴)^𝑝≤𝜋^𝑝_𝑝(𝐴)𝐴∈Π_𝑝;2≤𝑝<∞ (36) (see also Theorem  17.4.3 in [10, page 298]). Since 𝜋_𝑝(𝐴) is a norm, Theorems 1 and 4 imply the following.

Corollary 6. Let 𝐴,𝐵∈Π_𝑝 for some integer 𝑝≥2. Then |det_𝑝(𝐼−𝐴)|≤exp[𝛾_𝑝𝜋^𝑝_𝑝(𝐴)] and det_𝑝(𝐼−𝐴)−det_𝑝(𝐼−𝐵)≤𝜋_𝑝(𝐴−𝐵)exp𝛾_𝑝11+2𝜋_𝑝(𝐴−𝐵)+𝜋_𝑝(𝐴+𝐵)^𝑝. (37) If, in addition, (20) holds, then det_𝑝(𝐼−𝐴)≥𝑒^{−(𝑙𝜋𝑝𝑝(𝐴)/𝜙𝐴)}. (38)

Furthermore, let 𝐿^𝑝_𝜇(Ω)(Ω⊂ℝ^𝑛;1<𝑝<∞) be the space of scalar functions 𝑓 defined on Ω with a finite positive measure 𝜇 and the norm 𝑓=_Ω𝑓(𝑥)^𝑝𝑑𝜇^1/𝑝. (39) Let 𝐾∶𝐿^𝑝_𝜇(Ω)→𝐿^𝑝_𝜇(Ω) be the integral operator 𝐾𝑓(𝑡)=_Ω𝑘(𝑡,𝑠)𝑓(𝑠)𝑑𝜇 (40) whose kernel 𝑘 defined on Ω×Ω satisfies the condition 𝑘_𝑝(𝐾)∶=_ΩΩ|𝑘(𝑡,𝑠)|^𝑝′𝑑𝜇(𝑠)^𝑝/𝑝′𝑑𝜇(𝑡)^1/𝑝<∞, (41) 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 𝜋_𝑝(𝐾)≤𝑘_𝑝(𝐾). (42) 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 𝐿^𝑝_𝜇(Ω) for an integer 𝑝≥2. Then |det_𝑝(𝐼−𝐾)|≤exp[𝛾_𝑝𝑘^𝑝_𝑝(𝐾)] and det_𝑝(𝐼−𝐾)−det_𝑝𝐾≤𝑘𝐼−_𝑝𝐾−𝐾exp𝛾_𝑝11+2𝑘_𝑝𝐾−𝐾+𝑘_𝑝𝐾+𝐾^𝑝. (43) If, in addition, condition (20) holds for 𝐴=𝐾, then det_𝑝(𝐼−𝐾)≥𝑒^−(𝑙𝑘^𝑝_{𝑝(𝐾)/𝜙𝐾)}. (44)

Now let us consider a linear operator 𝑇 in 𝑙^𝑝  (1<𝑝<∞) generated by an infinite matrix (𝑡_𝑗𝑘)^∞_𝑗,𝑘=1, satisfying 𝜏_𝑝(𝑇)∶=^∞_𝑗=1^∞_𝑘=1𝑡_𝑗𝑘^{𝑝′𝑝/𝑝′1/𝑝}<∞. (45) 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 𝜋_𝑝(𝑇)≤𝜏_𝑝(𝑇), (46) 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 |det_𝑝(𝐼−𝑇)|≤exp[𝛾_𝑝𝜏^𝑝_𝑝(𝑇)] and det_𝑝(𝐼−𝑇)−det_𝑝𝑇≤𝜏𝐼−_𝑝𝛾𝑇−𝑇×exp_𝑝11+2𝜏_𝑝𝑇−𝑇+𝜏_𝑝𝑇+𝑇^𝑝. (47) If, in addition, condition (20) holds for 𝐴=𝑇, then det_𝑝(𝐼−𝑇)≥𝑒^−(𝑙𝜏^𝑝_{𝑝(𝑇)/𝜙𝑇)}. (48)

Now let 𝑋=𝐻 be a separable Hilbert space and 𝐿_𝑞,𝑟  (𝑞>1,0<𝑟<𝑞) the Lorentz ideal of compact operators 𝑇 with the finite quasinorm 𝑁_𝑞,𝑟(𝑇)=^∞_𝑘=1𝑘^{(𝑞/𝑟)−1}𝑠^𝑞_𝑘(𝑇)^1/𝑞, (49) where 𝑠_𝑘(𝑇) are the singular numbers of 𝑇 taken with their multiplicities. So 𝑁_𝑞,𝑟𝑇+𝑇≤𝑐_𝑞,𝑟𝑁_𝑞,𝑟(𝑇)+𝑁_𝑞,𝑟𝑇𝑐_𝑞,𝑟=const;𝑇,𝑇∈𝐿_𝑞,𝑟. (50) For the details, see [11, Section 1.1]. By [11, Lemma 1.4], ^∞_𝑘=1𝑘^{(𝑞/𝑟)−1}𝜆_𝑘(𝑇)^𝑞≤𝑐_𝑞,𝑟𝑁^𝑞_𝑞,𝑟(𝑇). (51) 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 ^∞_𝑘=1𝜆_𝑘(𝑇)^𝑝≤𝜏(𝑑)^∞_𝑘=1𝑘^𝑑𝜆_𝑘(𝑇)^{𝑝𝑑1/𝑑} (52) with 𝜏(𝑑)=^∞_𝑘=1𝑘^{−𝑑′1/𝑑′}1𝑑^′+1𝑑=1. (53) So we have ^∞_𝑘=1𝜆_𝑘(𝑇)^𝑝≤𝜏𝑞/𝑝^∞_𝑘=1𝑘^{(𝑞/𝑟)−1}𝜆_𝑘(𝑇)^{𝑞𝑝/𝑞}. (54) Thus (51) implies the following result.

Lemma 9. For an integer 𝑝≥1 and a 𝑞>𝑝, let 𝑇∈𝐿_𝑞,𝑟 with 𝑟=𝑞𝑝/(𝑝+𝑞). Then ^∞_𝑘=1𝜆_𝑘(𝑇)^𝑝≤𝑐_𝑞,𝑟𝜏𝑞𝑝𝑁^𝑝_𝑞,𝑟(𝑇). (55)

Now we can directly apply Theorems 1 and 4.