Contractions of Product Density Operators of Systems of Identical Fermions and Bosons

Abstract

Recurrence and explicit formulae for contractions (partial traces) of antisymmetric and symmetric products of identical trace class operators are derived. Contractions of product density operators of systems of identical fermions and bosons are proved to be asymptotically equivalent to, respectively, antisymmetric and symmetric products of density operators of a single particle, multiplied by a normalization integer. The asymptotic equivalence relation is defined in terms of the thermodynamic limit of expectation values of observables in the states represented by given density operators. For some weaker relation of asymptotic equivalence, concerning the thermodynamic limit of expectation values of product observables, normalized antisymmetric and symmetric products of density operators of a single particle are shown to be equivalent to tensor products of density operators of a single particle.

1. Introduction

This paper (see also preprint [1]), presenting the results of a part of the author's thesis [2], deals with contractions (partial traces) of antisymmetric and symmetric product density operators representing mixed states of systems of identical noninteracting fermions and bosons, respectively.

If ℋ is a separable Hilbert space of a single fermion (boson), then the space of the 𝑛-fermion (resp. 𝑛-boson) system is the antisymmetric (resp. symmetric) subspace ℋ^∧𝑛 (resp. ℋ^∨𝑛) of ℋ^⊗𝑛. Density operators of 𝑛-fermion (resp. 𝑛-boson) systems are identified with those defined on ℋ^⊗𝑛 and concentrated on ℋ^∧𝑛 (resp. ℋ^∨𝑛).

Recall that the expectation value of an observable represented by a bounded self-adjoint operator 𝐵 on given Hilbert space in a state described by a density operator 𝜌 equals Tr𝐵𝜌. If 𝐵 is an unbounded self-adjoint operator on a dense subspace of given Hilbert space, instead of 𝐵 one can consider its spectral measure 𝐸_𝐵(Δ) (which is a bounded operator) of a Borel subset Δ of the spectrum of 𝐵. Then Tr𝐸_𝐵(Δ)𝜌 is the probability that the result of the measurement of the observable in question belongs to Δ [3].

𝑘-particle observables of 𝑛-fermion and 𝑛-boson systems (𝑘<𝑛) are represented, respectively, by operators of the formΓ^∧𝑛_𝑘𝐵=𝐴_ℋ^(𝑛)𝐵⊗𝐼^{⊗(𝑛−𝑘)}𝐴_ℋ^(𝑛),Γ^∨𝑛_𝑘𝐵=𝑆_ℋ^(𝑛)𝐵⊗𝐼^{⊗(𝑛−𝑘)}𝑆_ℋ^(𝑛) (1.1) (multiplied by (^𝑛_𝑘)), where 𝐴_ℋ^(𝑛) and 𝑆_ℋ^(𝑛) are projectors of ℋ^⊗𝑛 onto ℋ^∧𝑛 and ℋ^∨𝑛, respectively, 𝐼 is the identity operator on ℋ and 𝐵 is a self-adjoint operator on ℋ^⊗𝑘 (see [4]). Operators (1.1) are called antisymmetric and symmetric expansions of 𝐵. In view of the earlier remark it is assumed that 𝐵 is bounded. The expectation values of observables represented by Γ^∧𝑛_𝑘𝐵 and Γ^∨𝑛_𝑘𝐵 in states represented by 𝑛-fermion and 𝑛-boson density operators 𝐾 and 𝐺, respectively, can be expressed asTr𝐾Γ^∧𝑛_𝑘𝐵=Tr𝐵L^𝑘_𝑛𝐾,Tr𝐺Γ^∨𝑛_𝑘𝐵=Tr𝐵L^𝑘_𝑛𝐺 (1.2) (see [4, equations (1.7), (3.19)]), where 𝑘-particle density operators L^𝑘_𝑛𝐾 and L^𝑘_𝑛𝐺 are (𝑛,𝑘)-contractions of 𝐾 and 𝐺 (see Definition 2.1), called also reduced density operators. Such operators were investigated by Coleman [5], Garrod and Percus [6], and Kummer [4] (see also, e.g., [7–9] and references therein). A presentation of the basic ideas concerning reduced density operators and their applications can be found in [10].

In the present paper particular interest is taken in the case when 𝐾 and 𝐺 are product density operators, that is, they are of the form1𝐾=Tr𝜌^∧𝑛𝜌^∧𝑛1,𝐺=Tr𝜌^∨𝑛𝜌^∨𝑛, (1.3) where 𝜌^∧𝑛=𝐴_ℋ^(𝑛)𝜌^⊗𝑛𝐴_ℋ^(𝑛), 𝜌^∨𝑛=𝑆_ℋ^(𝑛)𝜌^⊗𝑛𝑆_ℋ^(𝑛), and 𝜌 is a density operator of a single fermion or boson, respectively. The first objective of this paper is to find the recurrence and explicit formulae for L^𝑘_𝑛𝐾 and L^𝑘_𝑛𝐺 for 𝐾 and 𝐺 being, respectively, antisymmetric and symmetric products of identical trace class operators, including operators (1.3). The explicit form of the operators L^𝑘_𝑛𝐾 and L^𝑘_𝑛𝐺 proves to be quite complex. However, they can be replaced by operators with simpler structure if only the limiting values of expectations (1.2), in the sense of the thermodynamic limit, are of interest. The second objective of this paper is to find that simpler forms of contractions L^𝑘_𝑛𝐾 and L^𝑘_𝑛𝐺 for product density operators (1.3), equivalent to the complete expressions in the thermodynamic limit.

The problems described above have been solved for 𝑘=1,2 by Kossakowski and Maćkowiak [11], and Maćkowiak [12]. The formulae they derived were exploited in calculations of the free energy density of large interacting 𝑛-fermion and 𝑛-boson systems [11, 12], as well as in the perturbation expansion of the free energy density for the 𝑀-impurity Kondo Hamiltonian [13]. In the case of investigation of many-particle interactions of higher order [14–17], or higher order perturbation expansion terms of the free energy density, the expressions for (Tr𝜌^∧𝑛)⁻¹L^𝑘_𝑛𝜌^∧𝑛 and (Tr𝜌^∨𝑛)⁻¹L^𝑘_𝑛𝜌^∨𝑛 with 𝑘≥3 can be used in the canonical and grand canonical ensemble approach, which is the physical motivation for the present paper.

The main results of this paper are Theorems 3.1, 3.4, 4.9, and 4.14.

2. Preliminaries

In this section notation and terminology are set up.

2.1. Basic Notation

Let (ℋ,⟨⋅,⋅⟩) be a separable Hilbert space over ℂ or ℝ. The following notation is used in the sequel. 𝐼: the identity operator on ℋ,ℬ(ℋ): the space of bounded linear operators on ℋ with the operator norm ‖⋅‖, 𝒯(ℋ): the space of trace class operators on ℋ with the trace norm Tr|⋅|,ℬ^∗(ℋ): the space of bounded self-adjoint operators on ℋ, ℬ^∗_≥0(ℋ): the set of nonnegative definite bounded self-adjoint operators on ℋ, 𝒟(ℋ): the set of density operators (matrices) on ℋ, that is, 𝒟(ℋ)=𝐷∈𝒯(ℋ)∣𝐷=𝐷^∗,𝐷≥0,Tr𝐷=1. (2.1)

Set ℋ^⊗𝑛=ℋ⊗⋯⊗ℋ_𝑛 and denote by 𝑆_𝑛 the group of permutations of the set {1,…,𝑛}. Let 𝐴_ℋ^(𝑛),𝑆_ℋ^(𝑛)∈ℬ(ℋ^⊗𝑛) be the projectors such that 𝐴_ℋ^(𝑛)𝜓₁⊗⋯⊗𝜓_𝑛=1𝑛!_{𝜋∈𝑆𝑛}sgn𝜋𝜓_𝜋(1)⊗⋯⊗𝜓_𝜋(𝑛),𝑆_ℋ^(𝑛)𝜓₁⊗⋯⊗𝜓_𝑛=1𝑛!_{𝜋∈𝑆𝑛}𝜓_𝜋(1)⊗⋯⊗𝜓_𝜋(𝑛) (2.2) for every 𝜓₁,…,𝜓_𝑛∈ℋ. The closed linear subspaces ℋ^∧𝑛=𝐴_ℋ^(𝑛)ℋ^⊗𝑛 and ℋ^∨𝑛=𝑆_ℋ^(𝑛)ℋ^⊗𝑛 of ℋ^⊗𝑛 are called the antisymmetric and symmetric subspace, respectively.

The antisymmetric and symmetric product of operators 𝐵∈ℬ(ℋ^⊗𝑘), 𝐶∈ℬ(ℋ^⊗𝑚) are defined as 𝐵∧𝐶=𝐴_ℋ^(𝑘+𝑚)(𝐵⊗𝐶)𝐴_ℋ^(𝑘+𝑚) and 𝐵∨𝐶=𝑆_ℋ^(𝑘+𝑚)(𝐵⊗𝐶)𝑆_ℋ^(𝑘+𝑚), respectively. It is assumed 𝐵^∧𝑛=𝐵∧⋯∧𝐵_𝑛, 𝐵^∨𝑛=𝐵∨…∨𝐵_𝑛, and 𝐵^∧1=𝐵^∨1=𝐵. Clearly, if 𝐵∈ℬ(ℋ) then 𝐵^∧𝑛=𝐴_ℋ^(𝑛)𝐵^⊗𝑛=𝐵^⊗𝑛𝐴_ℋ^(𝑛), 𝐵^∨𝑛=𝑆_ℋ^(𝑛)𝐵^⊗𝑛=𝐵^⊗𝑛𝑆_ℋ^(𝑛), and if 𝐵∈ℬ^∗_≥0(ℋ) then 𝐵^∧𝑛,𝐵^∨𝑛∈ℬ^∗_≥0(ℋ^⊗𝑛).

Set ℝ₊=[0,+∞) and ℝ₊=ℝ₊∪{+∞}. The product of measures𝜇,𝜇₁ is denoted by 𝜇⊗𝜇₁ and 𝜇^⊗𝑛 stands for 𝜇⊗⋯⊗𝜇_𝑛. In subsequent sections use is made of product integral kernels, described in the appendix.

2.2. Contractions of Operators

The definition and basic properties of contractions of operators are now recalled for the reader's convenience. They were studied in [4–7]. A discussion of properties of reduced density operators can also be found in [10].

Let ℋ be a separable Hilbert space over the field 𝕂=ℂ or ℝ.

Definition 2.1. Let 𝑘,𝑛∈ℕ, 𝑘<𝑛, and 𝐾∈𝒯(ℋ^⊗𝑛). Then the (𝑛,𝑘)-contraction of 𝐾 is the operator L^𝑘_𝑛𝐾∈𝒯(ℋ^⊗𝑘) such that ∀_{𝐶∈ℬ(ℋ}^⊗𝑘₎∶Tr_ℋ^⊗𝑛𝐶⊗𝐼^{⊗(𝑛−𝑘)}𝐾=Tr_ℋ^⊗𝑘𝐶L^𝑘_𝑛𝐾. (2.3) It is also assumed L^𝑛_𝑛𝐾=𝐾.

Remark 2.2. The operator L^𝑘_𝑛𝐾 always exists and is defined uniquely by (2.3). L^𝑘_𝑛𝐾 is a partial trace of 𝐾 with respect to the component ℋ^{⊗(𝑛−𝑘)} of ℋ^⊗𝑛=ℋ^⊗𝑘⊗ℋ^{⊗(𝑛−𝑘)}. If ℋ=ℋ_𝑌∶=𝐿²(𝑌,𝜇), where the measure 𝜇is separable and 𝜎-finite, and 𝒦 is a product integral kernel of 𝐾 (see the appendix) then L^𝑘_𝑛𝐾 has an integral kernel 𝒦₀ given by formula (A.4), according to Lemma A.5 and Corollary A.6.

Under the assumptions of Definition 2.1 one has Tr_ℋ^⊗𝑘L^𝑘_𝑛𝐾=Tr_ℋ^⊗𝑛𝐾, and if 𝑝∈ℕ, 𝑘<𝑝<𝑛, then L^𝑘_𝑝(L^𝑝_𝑛𝐾)=L^𝑘_𝑛𝐾. Moreover, if 𝐾∈ℬ^∗(ℋ^⊗𝑛) then L^𝑘_𝑛𝐾∈ℬ^∗(ℋ^⊗𝑘), and if 𝐾∈ℬ^∗_≥0(ℋ^⊗𝑛) then L^𝑘_𝑛𝐾∈ℬ^∗_≥0(ℋ^⊗𝑘).

Contractions of density operators are called reduced density operators. Contractions preserve the Fermi and the Bose-Einstein statistics of the contracted operator, that is, for 𝐾∈𝐴_ℋ^(𝑛)𝒯(ℋ^⊗𝑛)𝐴_ℋ^(𝑛) and 𝐺∈𝑆_ℋ^(𝑛)𝒯(ℋ^⊗𝑛)𝑆_ℋ^(𝑛) one has L^𝑘_𝑛𝐾∈𝐴_ℋ^(𝑘)𝒯(ℋ^⊗𝑘)𝐴_ℋ^(𝑘) and L^𝑘_𝑛𝐺∈𝑆_ℋ^(𝑘)𝒯(ℋ^⊗𝑘)𝑆_ℋ^(𝑘). For such 𝐾 and 𝐺 (1.2) hold.

The following theorem is a part of Coleman's theorem [4, 5].

Theorem 2.3. Let 𝑛∈ℕ, 𝑛≥2. For every (𝑛-fermion) density operator 𝐷∈𝒟(ℋ^⊗𝑛), 𝐷=𝐴_ℋ^(𝑛)𝐷𝐴_ℋ^(𝑛), one has ‖L¹_𝑛𝐷‖≤(1/𝑛)‖𝐷‖.

3. Recurrence and Explicit Formulae for Contractions of Products of Trace Class Operators

In this section recurrence and explicit formulae for contractions of antisymmetric and symmetric powers of single particle operators are derived.

In the whole section use is made of the Hilbert space ℋ_𝑌∶=𝐿²(𝑌,𝜇) over the field 𝕂=ℂ or ℝ, where the measure μ is separable and 𝜎-finite.

The following theorem in the case of 𝑘=1,2 was proved in [11, 12].

Theorem 3.1 (Recurrence formulae). Let 𝜌∈𝒯(ℋ_𝑌). If 𝑘,𝑛∈ℕ, 1<𝑘<𝑛, then 𝑛𝑘L^𝑘_𝑛𝜌^∧𝑛=L𝑛−1𝑘−1^{𝑘−1𝑛−1}𝜌^{∧(𝑛−1)}−𝑘L∧𝜌𝑛−1_𝑘𝑛−1𝜌^{∧(𝑛−1)}𝐼^{⊗(𝑘−1)}⊗𝜌𝐴_ℋ^(𝑘)_𝑌,𝑛𝑘L^𝑘_𝑛𝜌^∨𝑛=L𝑛−1𝑘−1^{𝑘−1𝑛−1}𝜌^{∨(𝑛−1)}+𝑘L∨𝜌𝑛−1_𝑘𝑛−1𝜌^{∨(𝑛−1)}𝐼^{⊗(𝑘−1)}⊗𝜌𝑆_ℋ^(𝑘)_𝑌, (3.1)(3.2) and if 𝑛∈ℕ, 𝑛≥2, then 𝑛L¹_𝑛𝜌^∧𝑛=Tr𝜌^{∧(𝑛−1)}𝜌−(𝑛−1)L¹_𝑛−1𝜌^{∧(𝑛−1)}𝜌,𝑛L¹_𝑛𝜌^∨𝑛=Tr𝜌^{∨(𝑛−1)}𝜌+(𝑛−1)L¹_𝑛−1𝜌^{∨(𝑛−1)}𝜌. (3.3)(3.4)

Proof. Let 𝜚∶𝑌×𝑌→𝕂 be a product integral kernel of 𝜌. For every 𝑚∈ℕ define the mapping 𝜚^∧𝑚∶𝑌^𝑚×𝑌^𝑚→𝕂 by the formula 𝜚^∧𝑚𝑥₁,…,𝑥_𝑚𝑦₁,…,𝑦_𝑚𝜚𝑥=det¹,𝑦₁⋯𝜚𝑥₁,𝑦_𝑚𝜚𝑥⋮⋯⋮_𝑚,𝑦₁⋯𝜚𝑥_𝑚,𝑦_𝑚. (3.5) Then the mapping 𝒦∶𝑌^𝑛×𝑌^𝑛→𝕂 given by 𝒦𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛=1𝜚𝑛!^∧𝑛𝑥₁,…,𝑥_𝑛𝑦₁,…,𝑦_𝑛 (3.6) is a product integral kernel of 𝜌^∧𝑛=𝐴_ℋ^(𝑛)_𝑌𝜌^⊗𝑛.
Equation (3.1) will be first proved for 𝑛>𝑘+1. In view of Remark 2.2, an integral kernel ℒ∶𝑌^𝑘×𝑌^𝑘→𝕂 of (^𝑛_𝑘)L^𝑘_𝑛𝜌^∧𝑛 can be given byℒ𝑥^′,𝑦^′=𝑛𝑘^{𝑌𝑛−𝑘}𝒦𝑥^′,𝑥^′′,𝑦^′,𝑥^′′d𝜇^{⊗(𝑛−𝑘)}𝑥^′′ (3.7) for 𝜇^⊗2𝑘-a.a. (𝑥^′,𝑦^′)∈𝑌^𝑘×𝑌^𝑘. Performing 𝑘! permutations of the first 𝑘 rows and 𝑘! permutations of the first 𝑘 columns of the determinant defining 𝒦 and expanding that determinant with respect to the 𝑘th column one obtains ℒ𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘=𝑛𝑘11𝑛!(𝑘!)²_{𝜋,𝜎∈𝑆𝑘}sgn𝜋sgn𝜎^𝑘_𝑗=1(−1)^𝑘+𝑗×_{𝑌𝑛−𝑘}𝜚𝑥_𝜋(𝑗),𝑦_𝜎(𝑘)⋅𝜚^{∧(𝑛−1)}𝑥_𝜋(1),…,𝑥_{𝜋(𝑗−1)},𝑥_𝜋(𝑗+1),…,𝑥_𝜋(𝑘),𝑥_𝑘+1,…,𝑥_𝑛𝑦_𝜎(1),…,𝑦_{𝜎(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛×d𝜇^{⊗(𝑛−𝑘)}𝑥_𝑘+1,…,𝑥_𝑛+𝑛𝑘11𝑛!(𝑘!)²_{𝜋,𝜎∈𝑆𝑘}sgn𝜋sgn𝜎^𝑛_𝑗=𝑘+1(−1)^𝑘+𝑗×_{𝑌𝑛−𝑘}𝜚𝑥_𝑗,𝑦_𝜎(𝑘)⋅𝜚^{∧(𝑛−1)}𝑥_𝜋(1),…,𝑥_𝜋(𝑘),𝑥_𝑘+1,…,𝑥_𝑗−1,𝑥_𝑗+1,…,𝑥_𝑛𝑦_𝜎(1),…,𝑦_{𝜎(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛×d𝜇^{⊗(𝑛−𝑘)}𝑥_𝑘+1,…,𝑥_𝑛. (3.8)
Consider the first term on the r.h.s. of (3.8). In all summands of ∑^𝑘_𝑗=1 except the last one the (𝑘−1)th row of the determinant (containing the variable 𝑥_𝜋(𝑘)) can be shifted into the 𝑗th position, changing thereby the sign of the determinant by (−1)^{(𝑘−2)−(𝑗−1)}=(−1)^{−𝑘−𝑗+1}. Then the first term of sum (3.8) assumes the form𝑛𝑘11𝑛!(𝑘!)²_{𝜋,𝜎∈𝑆𝑘}sgn𝜋sgn𝜎^𝑘−1_𝑗=1(−1)^𝑘+𝑗(−1)^{−𝑘−𝑗+1}×_{𝑌𝑛−𝑘}𝜚𝑥_𝜋(𝑗),𝑦_𝜎(𝑘)⋅𝜚^{∧(𝑛−1)}𝑥_𝜋(1),…,𝑥_{𝜋(𝑗−1)},𝑥_𝜋(𝑘),𝑥_𝜋(𝑗+1),…,𝑥_{𝜋(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛𝑦_𝜎(1),…,𝑦_{𝜎(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛×d𝜇^{⊗(𝑛−𝑘)}𝑥_𝑘+1,…,𝑥_𝑛+𝑛𝑘11𝑛!(𝑘!)²_{𝜋,𝜎∈𝑆𝑘}sgn𝜋sgn𝜎(−1)^𝑘+𝑘×_{𝑌𝑛−𝑘}𝜚𝑥_𝜋(𝑘),𝑦_𝜎(𝑘)⋅𝜚^{∧(𝑛−1)}𝑥_𝜋(1),…,𝑥_{𝜋(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛𝑦_𝜎(1),…,𝑦_{𝜎(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛d𝜇^{⊗(𝑛−𝑘)}𝑥_𝑘+1,…,𝑥_𝑛. (3.9) Let 𝑇_𝑗𝑘∈𝑆_𝑘 denote the transposition 𝑗↔𝑘 for 𝑗<𝑘 (then (−1)^𝑘+𝑗(−1)^{−𝑘−𝑗+1}=(−1)=sgn𝑇_𝑗𝑘) and the identity permutation for 𝑗=𝑘 (with sgn𝑇_𝑘𝑘=1). Expression (3.9) can be written as ^𝑘_𝑗=1𝑛𝑘11𝑛!(𝑘!)²_{𝜋,𝜎∈𝑆𝑘}sgn𝜋sgn𝑇_𝑗𝑘sgn𝜎×_{𝑌𝑛−𝑘}𝜚𝑥_{(𝜋∘𝑇𝑗𝑘)(𝑘)},𝑦_𝜎(𝑘)⋅𝜚^{∧(𝑛−1)}𝑥_{(𝜋∘𝑇𝑗𝑘)(1)},…,𝑥_{(𝜋∘𝑇𝑗𝑘)(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛𝑦_𝜎(1),…,𝑦_{𝜎(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛×d𝜇^{⊗(𝑛−𝑘)}𝑥_𝑘+1,…,𝑥_𝑛=𝑛−11𝑘−1(𝑘!)²_{𝜏,𝜎∈𝑆𝑘}sgn𝜏sgn𝜎𝜚𝑥_𝜏(𝑘),𝑦_𝜎(𝑘)×_{𝑌𝑛−𝑘}1(𝑛−1)!⋅𝜚^{∧(𝑛−1)}𝑥_𝜏(1),…,𝑥_{𝜏(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛𝑦_𝜎(1),…,𝑦_{𝜎(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛d𝜇^{⊗(𝑛−𝑘)}𝑥_𝑘+1,…,𝑥_𝑛. (3.10) The function 𝒫₁∶𝑌^𝑘×𝑌^𝑘→𝕂, such that 𝒫₁(𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘) is 𝜇^⊗2𝑘-a.e. equal to expression (3.10), is an integral kernel of the operator L𝑛−1𝑘−1^{𝑘−1𝑛−1}𝜌^{∧(𝑛−1)}∧𝜌, (3.11) which appears on the r.h.s. of (3.1).
Consider now the second term of the sum on the r.h.s. of (3.8). One can change the indices of the integral variables 𝑥_𝑘+1,…,𝑥_𝑗 in all summands of ∑^𝑛_𝑗=𝑘+1 except the first one, according to the rule 𝑥_𝑗→𝑥_𝑘+1→𝑥_𝑘+2→⋯→𝑥_𝑗 for the 𝑗th summand, and simultaneously change the order of the columns of the determinant inversely (which changes the sign by (−1)^{(𝑗−1)−𝑘}=(−1)^{(𝑘+1)−𝑗}). The resulting sum ∑^𝑛_𝑗=𝑘+1 then contains 𝑛−𝑘 terms identical to the one with 𝑗=𝑘+1. Thus the second term of sum (3.8) equals−()𝑛𝑘1𝑛−𝑘1𝑛!(𝑘!)²_{𝜋,𝜎∈𝑆𝑘}sgn𝜋sgn𝜎×_{𝑌𝑛−𝑘}𝜚𝑥_𝑘+1,𝑦_𝜎(𝑘)⋅𝜚^{∧(𝑛−1)}𝑥_𝜋(1),…,𝑥_𝜋(𝑘),𝑥_𝑘+2,…,𝑥_𝑛𝑦_𝜎(1),…,𝑦_{𝜎(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛d𝜇^{⊗(𝑛−𝑘)}𝑥_𝑘+1,…,𝑥_𝑛.𝑘1=−𝑛−1𝑘!_{𝜎∈𝑆𝑘}×sgn𝜎_𝑌𝜚𝑥_𝑘+1,𝑦_{𝜎(𝑘)𝑌𝑛−1−𝑘}1(𝑛−1)!⋅𝜚^{∧(𝑛−1)}𝑥₁,…,𝑥_𝑘,𝑥_𝑘+2,…,𝑥_𝑛𝑦_𝜎(1),…,𝑦_{𝜎(𝑘−1)},𝑥_𝑘+1,…,𝑥_𝑛×d𝜇^{⊗(𝑛−1−𝑘)}𝑥_𝑘+2,…,𝑥_𝑛d𝜇𝑥_𝑘+1. (3.12) The function 𝒫₂∶𝑌^𝑘×𝑌^𝑘→𝕂, such that 𝒫₂(𝑥₁,…,𝑥_𝑘,𝑦₁,…,𝑦_𝑘) is 𝜇^⊗2𝑘-a.e. equal to expression (3.12), is an integral kernel of the operator −𝑘L𝑛−1_𝑘𝑛−1𝜌^{∧(𝑛−1)}𝐼^{⊗(𝑘−1)}⊗𝜌𝐴_ℋ^(𝑘)_𝑌, (3.13) which occurs on the r.h.s. of (3.1). One concludes that the kernel ℒ of the operator on the l.h.s. of (3.1) is 𝜇^⊗2𝑘-a.e. equal to the kernel 𝒫₁+𝒫₂ of the operator on the r.h.s. of (3.1), which proves the equality of both operators.
The proof of (3.1) for 𝑛=𝑘+1 and the proof of (3.3) proceed analogously.
Similarly, the proof of (3.2) and (3.4) is accomplished by changing the product ∧ into ∨ and replacing determinants in all formulae by pernaments, defined for every complex matrix 𝐴=[𝑎_𝑖,𝑗]^𝑚_𝑖,𝑗=1 as per𝐴=_{𝜋∈𝑆𝑚}𝑎_𝜋(1),1⋯𝑎_{𝜋(𝑚),𝑚}. (3.14) Notice that signs of permutations are omitted in this case, similarly as the multipliers ±1 in the Laplace expansions.

Lemma 3.2. Let 𝑘,𝑚∈ℕ, 1<𝑘<𝑚, 𝜌∈𝒯(ℋ_𝑌), 𝑗_𝑘∈{𝑘,…,𝑚}, and 𝑅∶=^𝑗𝑘−1_𝑗𝑘−1^𝑗_=𝑘−1^𝑘−1−1_{𝑗𝑘−2=𝑘−2}…^𝑗2−1_𝑗1=1𝜌^𝑗1⊗𝜌^{𝑗2−𝑗1}⊗⋯⊗𝜌^{𝑗𝑘−𝑗𝑘−1} (3.15) (for 𝑘=2 the only summation index is 𝑗_𝑘−1=𝑗₁). Then 𝐴_ℋ^(𝑘)_𝑌𝑅=𝑅𝐴_ℋ^(𝑘)_𝑌 and 𝑆_ℋ^(𝑘)_𝑌𝑅=𝑅𝑆_ℋ^(𝑘)_𝑌.

The proof of the above lemma consists in demonstrating the invariance of 𝑅 under permutations of factors in the tensor products. To this end it suffices to observe that 𝑅 is invariant under transpositions of neighbouring factors.

Lemma 3.3. Let 𝜌∈𝒯(ℋ_𝑌), 𝜉^∧_𝑠∶=Tr𝜌^∧𝑠, 𝜉^∨_𝑠∶=Tr𝜌^∨𝑠 for 𝑠∈ℕ, 𝜉^∧₀∶=1, 𝜉^∨₀∶=1, and Π_𝑚^∧𝑝𝜌∶=^𝑚_𝑗𝑝^𝑗_=𝑝^𝑝−1_{𝑗𝑝−1=𝑝−1}…^𝑗2−1_𝑗1=1𝜉^∧_{𝑚−𝑗𝑝}(−1)^{𝑝+𝑗𝑝}𝜌^𝑗1∧𝜌^{𝑗2−𝑗1}∧⋯∧𝜌^{𝑗𝑝−𝑗𝑝−1},Π_𝑚^∨𝑝𝜌∶=^𝑚_𝑗𝑝^𝑗_=𝑝^𝑝−1_{𝑗𝑝−1=𝑝−1}…^𝑗2−1_𝑗1=1𝜉^∨_{𝑚−𝑗𝑝}𝜌^𝑗1∨𝜌^{𝑗2−𝑗1}∨⋯∨𝜌^{𝑗𝑝−𝑗𝑝−1} (3.16) for 𝑝,𝑚∈ℕ, 𝑝≤𝑚. (For 𝑝=1 the only summation index is 𝑗₁ and the summation runs over the operators 𝜌^𝑗1.) If 2≤𝑝<𝑚 then Π_𝑚^∧𝑝𝜌=Π^{∧(𝑝−1)}_𝑚−1𝜌∧𝜌−Π^∧𝑝_𝑚−1𝜌𝐼^{⊗(𝑝−1)}⊗𝜌𝐴_ℋ^(𝑝)_𝑌Π_𝑚∨𝑝,𝜌=Π^{∨(𝑝−1)}_𝑚−1𝜌∨𝜌+Π^∨𝑝_𝑚−1𝜌𝐼^{⊗(𝑝−1)}⊗𝜌𝑆_ℋ^(𝑝)_𝑌. (3.17)(3.18)

Proof. Equation (3.17) will be first proved for 𝑝>2. One has Π_𝑚^∧𝑝𝜌=𝜉^∧_𝑚−𝑝+𝜌∧⋯∧𝜌_{𝑚−1𝑙𝑝𝑙=𝑝𝑝𝑙𝑝−1𝑙=𝑝−1𝑝−1−1𝑙𝑝−2=𝑝−2}…^𝑙2−1_𝑙1=1𝜉^∧_{𝑚−𝑙𝑝−1}(−1)^{𝑝+𝑙𝑝+1}𝜌^𝑙1∧𝜌^{𝑙2−𝑙1}∧⋯∧𝜌^{𝑙𝑝−1−𝑙𝑝−2}∧𝜌^{𝑙𝑝−𝑙𝑝−1+1}=𝜉^∧_𝑚−𝑝+𝜌∧⋯∧𝜌_{𝑚−1𝑙𝑝𝑙=𝑝𝑝−1𝑙𝑝−1𝑙=𝑝−1𝑝−1−1𝑙𝑝−2=𝑝−2}…^𝑙2−1_𝑙1=1𝜉^∧_{𝑚−𝑙𝑝−1}(−1)^{𝑝+𝑙𝑝+1}𝜌^𝑙1∧𝜌^{𝑙2−𝑙1}∧⋯∧𝜌^{𝑙𝑝−1−𝑙𝑝−2}∧𝜌^{𝑙𝑝−𝑙𝑝−1+1}+^𝑚−1_𝑙𝑝^𝑙_=𝑝^𝑝−1_𝑙𝑝−2^𝑙_=𝑝−2^𝑝−2−1_{𝑙𝑝−3=𝑝−3}…^𝑙2−1_𝑙1=1𝜉^∧_{𝑚−𝑙𝑝−1}(−1)^{𝑝+𝑙𝑝+1}𝜌^𝑙1∧𝜌^{𝑙2−𝑙1}∧⋯∧𝜌^{𝑙𝑝−2−𝑙𝑝−3}∧𝜌^{𝑙𝑝−𝑙𝑝−2}∧𝜌. (3.19)(3.20) The first and the third term after the last of equalities (3.20) yield ^𝑚−1_𝑗𝑝−1^𝑗_=𝑝−1^𝑝−1−1_{𝑗𝑝−2=𝑝−2}…^𝑗2−1_𝑗1=1𝜉^∧_{(𝑚−1)−𝑗𝑝−1}(−1)^{(𝑝−1)+𝑗𝑝−1}𝜌^𝑗1∧𝜌^{𝑗2−𝑗1}∧⋯∧𝜌^{𝑗𝑝−1−𝑗𝑝−2}∧𝜌 (3.21) for 𝑙_𝑝≡𝑗_𝑝−1, 𝑙_𝑝−2≡𝑗_𝑝−2,…, 𝑙₁≡𝑗₁. By Lemma 3.2, the second term after the last of equalities (3.20) equals −Π^∧𝑝_𝑚−1𝜌𝐼^{⊗(𝑝−1)}⊗𝜌𝐴_ℋ^(𝑝)_𝑌. (3.22) The sum of expressions (3.21) and (3.22) is equal to the r.h.s. of (3.17) for 𝑝>2. After simplifications the proof also applies to the case of 𝑝=2.
The proof of (3.18) is analogous to that of (3.17).

The next theorem provides the explicit form of (𝑛,𝑘)-contractions of product operators. The proof for 𝑘=1,2 was given in [11, 12]. The author of [12] emphasized that formula (3.23) for 𝑘=2 was derived by S. Pruski in 1978.

Theorem 3.4 (Explicit formulae). Let 𝑘,𝑛∈ℕ, 𝑘<𝑛, 𝜌∈𝒯(ℋ_𝑌), 𝜉^∧_𝑠∶=Tr𝜌^∧𝑠, 𝜉^∨_𝑠∶=Tr𝜌^∨𝑠 for 𝑠∈ℕ, and 𝜉^∧₀∶=1, 𝜉^∨₀∶=1. Then 𝑛𝑘L^𝑘_𝑛𝜌^∧𝑛=^𝑛_𝑗𝑘^𝑗_=𝑘^𝑘−1_{𝑗𝑘−1=𝑘−1}…^𝑗2−1_𝑗1=1𝜉^∧_{𝑛−𝑗𝑘}(−1)^{𝑘+𝑗𝑘}𝜌^𝑗1∧𝜌^{𝑗2−𝑗1}∧⋯∧𝜌^{𝑗𝑘−𝑗𝑘−1}=^{𝑛−(𝑘−1)}_𝑖1=1^{𝑛−𝑖1−(𝑘−2)}_𝑖2=1…^{𝑛−𝑖1−⋯−𝑖𝑘−2−1}_{𝑖𝑘−1=1}^{𝑛−𝑖1−⋯−𝑖𝑘−1}_𝑖𝑘=1𝜉^∧_{𝑛−𝑖1−⋯−𝑖𝑘}(−1)^{𝑘+𝑖1+⋯+𝑖𝑘}𝜌^𝑖1∧⋯∧𝜌^𝑖𝑘,𝑛𝑘L^𝑘_𝑛𝜌^∨𝑛=^𝑛_𝑗𝑘^𝑗_=𝑘^𝑘−1_{𝑗𝑘−1=𝑘−1}…^𝑗2−1_𝑗1=1𝜉^∨_{𝑛−𝑗𝑘}𝜌^𝑗1∨𝜌^{𝑗2−𝑗1}∨⋯∨𝜌^{𝑗𝑘−𝑗𝑘−1}=^{𝑛−(𝑘−1)}_𝑖1=1^{𝑛−𝑖1−(𝑘−2)}_𝑖2=1…^{𝑛−𝑖1−⋯−𝑖𝑘−2−1}_{𝑖𝑘−1=1}^{𝑛−𝑖1−⋯−𝑖𝑘−1}_𝑖𝑘=1𝜉^∨_{𝑛−𝑖1−⋯−𝑖𝑘}𝜌^𝑖1∨⋯∨𝜌^𝑖𝑘. (3.23)(3.24) (For 𝑘=1 the only summation indices are 𝑗₁ and 𝑖₁ and the summation runs over the operators 𝜌^𝑗1 and 𝜌^𝑖1, resp.)

Proof. For every 𝑝,𝑚∈ℕ, 𝑝≤𝑚, let Π_𝑚^∧𝑝(𝜌) be defined as in Lemma 3.3. Then the first of equalities (3.23) can be written as 𝑛𝑘L^𝑘_𝑛𝜌^∧𝑛=Π_𝑛^∧𝑘𝜌. (3.25) The proof of (3.25) will be carried out by (double) induction with respect to 𝑘 and, for fixed 𝑘, with respect to 𝑛>𝑘.(1^∘) (𝑘=1) This part of the proof is by induction with respect to 𝑛>1.(a)(𝑛=2) According to Theorem 3.1, 2L¹₂𝜌^∧2=(Tr𝜌)𝜌−𝜌²=Π₂^∧1(𝜌).(b)Assuming validity of formula (3.25) (with 𝑘=1) for 𝑛∈{2,…,𝑚−1}, where 𝑚∈ℕ, 𝑚>2, its validity will be shown for 𝑛=𝑚.
One hasΠ_𝑚^∧1𝜌=𝜉^∧_𝑚−1𝜌+^𝑚_𝑗1=2𝜉^∧_𝑚−𝑗1(−1)^1+𝑗1𝜌^𝑗1=𝜉^∧_𝑚−1𝜌−Π^∧1_𝑚−1𝜌𝜌. (3.26) Thus, according to the inductive hypothesis for 𝑛∈{2,…,𝑚−1}, Π_𝑚^∧1𝜌=𝜉^∧_𝑚−1𝜌−(𝑚−1)L¹_𝑚−1𝜌^{∧(𝑚−1)}𝜌, (3.27) which, in view of Theorem 3.1, yields (^𝑚₁)L¹_𝑚𝜌^∧𝑚=Π_𝑚^∧1(𝜌).(2^∘)Assuming validity of formula (3.25) for 𝑘∈{1,…,𝑝−1} (and every 𝑛>𝑘), where 𝑝∈ℕ, 𝑝>1, its validity will be shown for 𝑘=𝑝. For arbitrarily fixed 𝑝 the proof will be carried out by induction with respect to 𝑛>𝑝.(a)(𝑛=𝑝+1) By the inductive hypothesis with respect to 𝑘 and Lemma 3.3, Π^∧𝑝_𝑝+1𝜌=L𝑝+1−1𝑝−1^{𝑝−1(𝑝+1)−1}𝜌^{∧((𝑝+1)−1)}∧𝜌−𝜌^∧𝑝𝐼^{⊗(𝑝−1)}⊗𝜌𝐴_ℋ^(𝑝)_𝑌, (3.28) hence (_𝑝^𝑝+1)L^𝑝_𝑝+1𝜌^∧(𝑝+1)=Π^∧𝑝_𝑝+1(𝜌), according to Theorem 3.1.(b)Assuming validity of formula (3.25) for 𝑛∈{𝑝+1,…,𝑚−1}, where 𝑘=𝑝, 𝑚∈ℕ, 𝑚>𝑝+1, its validity will be shown for 𝑛=𝑚.
By the inductive hypothesis for 𝑘∈{1,…,𝑝−1} and Lemma 3.3 one hasΠ_𝑚^∧𝑝𝜌=L𝑚−1𝑝−1^{𝑝−1𝑚−1}𝜌^{∧(𝑚−1)}∧𝜌−Π^∧𝑝_𝑚−1𝜌𝐼^{⊗(𝑝−1)}⊗𝜌𝐴_ℋ^(𝑝)_𝑌. (3.29) According to the inductive hypothesis for 𝑛∈{𝑝+1,…,𝑚−1} one thus obtains Π_𝑚^∧𝑝𝜌=L𝑚−1𝑝−1^{𝑝−1𝑚−1}𝜌^{∧(𝑚−1)}𝑝∧𝜌−L𝑚−1_𝑝𝑚−1𝜌^{∧(𝑚−1)}𝐼^{⊗(𝑝−1)}⊗𝜌𝐴_ℋ^(𝑝)_𝑌, (3.30) which, in view of Theorem 3.1, yields (^𝑚_𝑝)L^𝑝_𝑚𝜌^∧𝑚=Π_𝑚^∧𝑝(𝜌). This completes the inductive proof for (3.25) with respect to 𝑛>𝑝 and with respect to 𝑘.
Now turn to the second of equalities (3.23). For 𝑘=1 it is identity. Let 𝑘≥2. Setting 𝑗₁=𝑖₁, 𝑗₂=𝑖₁+𝑖₂,…, 𝑗_𝑘=𝑖₁+⋯+𝑖_𝑘 or, equivalently, 𝑖₁=𝑗₁, 𝑖₂=𝑗₂−𝑗₁, 𝑖₃=𝑗₃−𝑗₂,…, 𝑖_𝑘=𝑗_𝑘−𝑗_𝑘−1, one checks that both sides of the equality in question are equal to^{𝑛−(𝑘−1)}_𝑗1=1^{𝑛−(𝑘−2)}_{𝑗2=𝑗1+1}…^𝑛−1_{𝑗𝑘−1=𝑗𝑘−2}^𝑛_{+1𝑗𝑘=𝑗𝑘−1+1}𝜉^∧_{𝑛−𝑗𝑘}(−1)^{𝑘+𝑗𝑘}𝜌^𝑗1∧𝜌^{𝑗2−𝑗1}∧⋯∧𝜌^{𝑗𝑘−𝑗𝑘−1}. (3.31)
The proof of (3.24) is analogous to that of (3.23).

4. Asymptotic Form for Contractions of Product States

The explicit forms of the contractions of product states given by Theorem 3.4 are quite complex. In the present section they are replaced by simpler operators, equivalent in the thermodynamic limit. The main results in this section are Theorems 4.9 and 4.14.

In what follows use is made of the Hilbert space ℋ_Ω∶=𝐿²(Ω,𝜇) (over ℂ or ℝ), where the measure𝜇 is separable, 𝜎-finite, and satisfies the condition 𝜇(Ω)=+∞. For every𝜇-measurable subset 𝑌⊂Ω it is assumed ℋ_𝑌∶=𝐿²(𝑌,𝜇).

Let ℳ(Ω) be a fixed family of measurable subsets of Ω such that 0<𝜇(𝑌)<+∞ for every 𝑌∈ℳ(Ω) (it can be the family of all such subsets). Fix 𝑑∈ℝ, 𝑑>0, and assume that there exists a sequence {𝑌_𝑛}_𝑛∈ℕ⊂ℳ(Ω) such that 𝑛/𝜇(𝑌_𝑛)→𝑑 as 𝑛→∞.

Definition 4.1. Fix 𝑑∈ℝ, 𝑑>0, and let {𝑏_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} be a family of complex numbers. A complex number 𝑏 is said to be the thermodynamic limit of this family if for every sequence {𝑌_𝑛}_𝑛∈ℕ⊂ℳ(Ω) such that lim_𝑛→∞𝑛/𝜇(𝑌_𝑛)=𝑑 the condition lim_𝑛→∞𝑏_{𝑌𝑛,𝑛}=𝑏 is fulfilled. In such a case 𝑏 is denoted by 𝑑−lim_{𝑛,𝜇(𝑌)→∞}𝑏_𝑌,𝑛.

Special attention will be given to the families of complex numbers of the form Tr(L^𝑘_𝑛𝐾_𝑌,𝑛)𝐶_𝑌, where 𝑘,𝑛∈ℕ, 𝑛>𝑘, 𝐾_𝑌,𝑛∈𝒯(ℋ_𝑌^⊗𝑛), and 𝐶_𝑌∈ℬ(ℋ_𝑌^⊗𝑘).

Definition 4.1 does not guarantee the convergence of families {𝑏_𝑌,𝑛} of interest in physics. To obtain such a convergence, additional conditions (such as conditions of uniform growth [18]) are usually imposed on the sequence {𝑌_𝑛}_𝑛∈ℕ in question. However, those additional conditions do not affect considerations in this paper.

Expression of expectation values of observables in mixed states by using trace, mentioned in Introduction, is the motivation for the following definition.

Definition 4.2. Fix 𝑘∈ℕ and 𝑑∈ℝ, 𝑑>0. Families {𝐴_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} and {𝐵_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} of operators 𝐴_𝑌,𝑛,𝐵_𝑌,𝑛∈𝒯(ℋ_𝑌^⊗𝑘) are said to be asymptotically equivalent (symbolically: 𝐴_𝑌,𝑛≈𝐵_𝑌,𝑛), if for every family {𝐶_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} of operators 𝐶_𝑌,𝑛∈ℬ(ℋ_𝑌^⊗𝑘) with uniformly bounded operator norms one has 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐴_𝑌,𝑛−𝐵_𝑌,𝑛𝐶_𝑌,𝑛=0. (4.1)

Condition (4.1) is required to hold in particular for families {𝐶_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} such that 𝐶_𝑌,𝑛=𝐶_𝑌,𝑚 for all 𝑌∈ℳ(Ω), 𝑛,𝑚∈ℕ.

Remark 4.3. The authors of [11, 12] used some different definition of asymptotic equivalence of families of operators, closer to Definition 4.10 in this paper.

Remark 4.4. For fixed 𝑘∈ℕ and 𝑑∈ℝ, 𝑑>0, the relation ≈ is an equivalence relation. If 𝐴_𝑌,𝑛≈𝐵_𝑌,𝑛 then for every family of operators 𝐶_𝑌,𝑛 as in Definition 4.2 the limit 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐴_𝑌,𝑛𝐶_𝑌,𝑛 exists if and only if the limit 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐵_𝑌,𝑛𝐶_𝑌,𝑛 exists, in which case both limits are equal. Notice also that if 𝐴_𝑌,𝑛≈𝐵_𝑌,𝑛 then 𝐴_𝑌,𝑛+𝐷_𝑌,𝑛≈𝐵_𝑌,𝑛+𝐷_𝑌,𝑛 and 𝑎𝐴_𝑌,𝑛≈𝑎𝐵_𝑌,𝑛 for every family {𝐷_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ}⊂𝒯(ℋ_𝑌^⊗𝑘) and 𝑎∈ℂ. Furthermore, for every family {𝐴_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ}⊂𝒯(ℋ_𝑌^⊗𝑘) with uniformly bounded trace norms Tr|𝐴_𝑌,𝑛| and for every sequence {𝑎_𝑛}_𝑛∈ℕ⊂ℂ convergent to 𝑎∈ℂ one has 𝑎_𝑛𝐴_𝑌,𝑛≈𝑎𝐴_𝑌,𝑛.

Lemma 4.5. Let {𝐴_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} and {𝐵_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} be as in Definition 4.2. Then 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐴_𝑌,𝑛−𝐵_𝑌,𝑛=0⟹𝐴_𝑌,𝑛≈𝐵_𝑌,𝑛. (4.2) Moreover, if the operators 𝐴_𝑌,𝑛, 𝐵_𝑌,𝑛 are self-adjoint then 𝐴_𝑌,𝑛≈𝐵_𝑌,𝑛⟹𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐴_𝑌,𝑛−𝐵_𝑌,𝑛=0. (4.3)

Proof. Implication (4.2) follows from Definition 4.2 and the estimate Tr𝐴_𝑌,𝑛−𝐵_𝑌,𝑛𝐶_𝑌,𝑛≤𝐶_𝑌,𝑛Tr𝐴_𝑌,𝑛−𝐵_𝑌,𝑛. (4.4)
Now assume that 𝐴_𝑌,𝑛≈𝐵_𝑌,𝑛, which is equivalent to the condition𝐷_𝑌,𝑛≈0, (4.5) where 𝐷_𝑌,𝑛∶=𝐴_𝑌,𝑛−𝐵_𝑌,𝑛. The operators 𝐷_𝑌,𝑛 have the spectral representations 𝐷_𝑌,𝑛=^∞_𝑖=1𝜆_𝑖(𝑌,𝑛)𝑃_{𝜑𝑖(𝑌,𝑛)}, (4.6) where 𝑃_{𝜑𝑖(𝑌,𝑛)} are the projectors onto orthogonal one dimensional subspaces of eigenvectors 𝜑_𝑖(𝑌,𝑛) of 𝐷_𝑌,𝑛, corresponding to eigenvalues 𝜆_𝑖(𝑌,𝑛)∈ℝ. Since ∑^∞_𝑖=1|𝜆_𝑖(𝑌,𝑛)|=Tr|𝐷_𝑌,𝑛|<+∞, for every (𝑌,𝑛)∈ℳ(Ω)×ℕ there exists 𝑚(𝑌,𝑛)∈ℕ such that ∑^∞_{𝑖=𝑚(𝑌,𝑛)+1}|𝜆_𝑖(𝑌,𝑛)|<1/𝑛. Thus the operators 𝐹_𝑌,𝑛=^{𝑚(𝑌,𝑛)}_𝑖=1𝜆_𝑖(𝑌,𝑛)𝑃_{𝜑𝑖(𝑌,𝑛)} (4.7) satisfy the condition 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐷_𝑌,𝑛−𝐹_𝑌,𝑛=𝑑−lim^∞_{𝑛,𝜇(𝑌)→∞𝑖=𝑚(𝑌,𝑛)+1}𝜆_𝑖(𝑌,𝑛)=0, (4.8) which, in view of implication (4.2) proved and condition (4.5), yields 𝐹_𝑌,𝑛≈𝐷_𝑌,𝑛≈0. In particular, 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐹_𝑌,𝑛𝐶_𝑌,𝑛=0, (4.9) where 𝐶_𝑌,𝑛=^{𝑚(𝑌,𝑛)}_𝑖=1sgn𝜆_𝑖(𝑌,𝑛)𝑃_{𝜑𝑖(𝑌,𝑛)},𝐶_𝑌,𝑛=1. (4.10) Observe that Tr𝐹_𝑌,𝑛𝐶_𝑌,𝑛=Tr|𝐹_𝑌,𝑛|, hence condition (4.9) gives 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐹_𝑌,𝑛=0. (4.11) Since Tr|𝐷_𝑌,𝑛|≤Tr|𝐷_𝑌,𝑛−𝐹_𝑌,𝑛|+Tr|𝐹_𝑌,𝑛|, conditions (4.8) and (4.11) yield 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐴_𝑌,𝑛−𝐵_𝑌,𝑛≡𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝐷_𝑌,𝑛=0, (4.12) which proves implication (4.3).

The following lemma follows from Lemma 4.5.

Lemma 4.6. Fix 𝑘,𝑚∈ℕ. Let {𝐴_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} and {𝐵_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} be families of self-adjoint operators 𝐴_𝑌,𝑛,𝐵_𝑌,𝑛∈𝒯(ℋ_𝑌^⊗𝑘) such that 𝐴_𝑌,𝑛≈𝐵_𝑌,𝑛, and let {𝐷_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} be a family of operators 𝐷_𝑌,𝑛∈𝒯(ℋ_𝑌^⊗𝑚) with uniformly bounded trace norms Tr|𝐷_𝑌,𝑛|. Then 𝐴_𝑌,𝑛⊗𝐷_𝑌,𝑛≈𝐵_𝑌,𝑛⊗𝐷_𝑌,𝑛,𝐷_𝑌,𝑛⊗𝐴_𝑌,𝑛≈𝐷_𝑌,𝑛⊗𝐵_𝑌,𝑛,𝐴_𝑌,𝑛∧𝐷_𝑌,𝑛≈𝐵_𝑌,𝑛∧𝐷_𝑌,𝑛,𝐷_𝑌,𝑛∧𝐴_𝑌,𝑛≈𝐷_𝑌,𝑛∧𝐵_𝑌,𝑛,𝐴_𝑌,𝑛∨𝐷_𝑌,𝑛≈𝐵_𝑌,𝑛∨𝐷_𝑌,𝑛,𝐷_𝑌,𝑛∨𝐴_𝑌,𝑛≈𝐷_𝑌,𝑛∨𝐵_𝑌,𝑛. (4.13)

In the sequel {𝜌_𝑌}_{𝑌∈ℳ(Ω)} denotes a family of nonnegative definite self-adjoint operators 𝜌_𝑌∈𝒯(ℋ_𝑌), and for every (𝑌,𝑛)∈ℳ(Ω)×ℕ it is assumed that 𝜉^∧_𝑌,0∶=1,𝜉^∨_𝑌,0∶=1,𝜌_𝑌^∧1∶=𝜌_𝑌,𝜌_𝑌^∨1∶=𝜌_𝑌,𝜉^∧_𝑌,𝑛∶=Tr𝜌_𝑌^∧𝑛>0,𝜉^∨_𝑌,𝑛∶=Tr𝜌_𝑌^∨𝑛𝑠>0,_∧𝑌,𝑛𝜉∶=^{∧𝑌,𝑛−1}𝜉^∧_𝑌,𝑛,𝑠^∨_𝑌,𝑛𝜉∶=^{∨𝑌,𝑛−1}𝜉^∨_𝑌,𝑛. (4.14)

The objective of this section is to find density operators of the most simple form which are asymptotically equivalent to the operators 𝜎^∧(𝑘)_𝑌,𝑛∶=L^𝑘_𝑛1𝜉^∧_𝑌,𝑛𝜌_𝑌^∧𝑛,𝜎^∨(𝑘)_𝑌,𝑛∶=L^𝑘_𝑛1𝜉^∨_𝑌,𝑛𝜌_𝑌^∨𝑛, (4.15) defined for fixed 𝑘∈ℕ and every (𝑌,𝑛)∈ℳ(Ω)×ℕ, 𝑛>𝑘.

Remark 4.7. For every (𝑌,𝑛)∈ℳ(Ω)×ℕ the operator 𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌 is invertible and ‖(𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌)⁻¹‖=1. Furthermore, if 𝑠^∨_𝑌,𝑛+1‖𝜌_𝑌‖<1 then 𝐼−𝑠^∨_𝑌,𝑛+1𝜌_𝑌 is invertible and ‖(𝐼−𝑠^∨_𝑌,𝑛+1𝜌_𝑌)⁻¹‖=(1−𝑠^∨_𝑌,𝑛+1‖𝜌_𝑌‖)⁻¹.

The next theorem is a version of a theorem studied in [11, 12] (see Remark 4.3).

Theorem 4.8. If 𝜎^∧(1)_𝑌,𝑛≈𝜎^∧(1)_𝑌,𝑛+1 and the reals 𝑠^∧_𝑌,𝑛+1‖𝜌_𝑌‖, (𝑌,𝑛)∈ℳ(Ω)×ℕ, are uniformly bounded then 𝜎^∧(1)_𝑌,𝑛𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌≈(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌,_∧𝜎⁽¹⁾_𝑌,𝑛≈(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌⁻¹. (4.16)(4.17) If 𝜎^∨(1)_𝑌,𝑛≈𝜎^∨(1)_𝑌,𝑛+1 and the reals 𝑠^∨_𝑌,𝑛‖𝜌_𝑌‖, (𝑌,𝑛)∈ℳ(Ω)×ℕ, are uniformly bounded then 𝜎^∨(1)_𝑌,𝑛𝐼−𝑠^∨_𝑌,𝑛+1𝜌_𝑌≈(𝑛+1)⁻¹𝑠^∨_𝑌,𝑛+1𝜌_𝑌. (4.18) If, additionally, 𝑠^∨_𝑌,𝑛‖𝜌_𝑌‖≤𝜖 for some 𝜖<1 and every (𝑌,𝑛)∈ℳ(Ω)×ℕ then 𝜎^∨(1)_𝑌,𝑛≈(𝑛+1)⁻¹𝑠^∨_𝑌,𝑛+1𝜌_𝑌𝐼−𝑠^∨_𝑌,𝑛+1𝜌_𝑌⁻¹. (4.19)

Proof. By Theorem 3.1 and the assumption 𝜎^∧(1)_𝑌,𝑛≈𝜎^∧(1)_𝑌,𝑛+1 one has 𝜎^∧(1)_𝑌,𝑛−(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌≈−(𝑛+1)⁻¹𝑛𝜎^∧(1)_𝑌,𝑛𝑠^∧_𝑌,𝑛+1𝜌_𝑌. (4.20) Since Tr|𝜎^∧(1)_𝑌,𝑛(𝑠^∧_𝑌,𝑛+1𝜌_𝑌)|≤𝑠^∧_𝑌,𝑛+1‖𝜌_𝑌‖Tr|𝜎^∧(1)_𝑌,𝑛|=𝑠^∧_𝑌,𝑛+1‖𝜌_𝑌‖, relation (4.20) yields (4.16), in view of Remark 4.4.
Now turn to the proof of relation (4.17). According to Remark 4.7,Tr𝜎^∧(1)_𝑌,𝑛−(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌⁻¹≤𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌⁻¹Tr𝜎^∧(1)_𝑌,𝑛𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌−(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌=Tr𝜎^∧(1)_𝑌,𝑛𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌−(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌. (4.21) The explicit form of 𝜎^∧(1)_𝑌,𝑛 given by Theorem 3.4 shows that 𝜎^∧(1)_𝑌,𝑛 commutes with 𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌, and since both operators are self-adjoint, 𝜎^∧(1)_𝑌,𝑛(𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌) is also self-adjoint. Thus conditions (4.16), (4.21), and Lemma 4.5 yield (4.17).
The proof of relations (4.18), (4.19) runs parallel to that of (4.16), (4.17). Notice that in this case the expression ‖(𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌)⁻¹‖=1 from estimate (4.21) is replaced by ‖(𝐼−𝑠^∧_𝑌,𝑛+1𝜌_𝑌)⁻¹‖=(1−𝑠^∧_𝑌,𝑛+1‖𝜌_𝑌‖)⁻¹≤(1−𝜖)⁻¹ (see Remark 4.7).

The following theorem for 𝑘=2 (with the reservation of Remark 4.3) was obtained in [11, 12]. The author of [12] gave also arguments that can be used to check the assumptions of this theorem.

Theorem 4.9 (Asymptotic formulae I). If 𝜎^∧(𝑘)_𝑌,𝑛≈𝜎^∧(𝑘)_𝑌,𝑛+1 for every 𝑘∈ℕ and 𝑠^∧_𝑌,𝑛𝜌_𝑌≤2forevery(Y,n)∈ℳ(Ω)×ℕ (4.22) then, for every 𝑘∈ℕ, 𝑘≥2, 𝜎^∧(𝑘)_𝑌,𝑛≈𝑘!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_{𝑌,𝑛𝑘}. (4.23)
If 𝜎^∨(𝑘)_𝑌,𝑛≈𝜎^∨(𝑘)_𝑌,𝑛+1 for every 𝑘∈ℕ and𝑠^∨_𝑌,𝑛𝜌_𝑌≤𝜖forsome𝜖<1andevery(𝑌,𝑛)∈ℳ(Ω)×ℕ (4.24) then, for every 𝑘∈ℕ, 𝑘≥2, 𝜎^∨(𝑘)_𝑌,𝑛≈𝑘!𝜎^∨(1)_𝑌,𝑛∨⋯∨𝜎^∨(1)_{𝑌,𝑛𝑘}. (4.25)

Proof. First equivalence (4.23) will be proved. Observe that 2Tr𝜎^∧(𝑞)_𝑌,𝑛−𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛=Tr𝜎^∧(𝑞)_𝑌,𝑛−𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌+𝜎^∧(𝑞)_𝑌,𝑛−𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝐼−𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌≤Tr𝜎^∧(𝑞)_𝑌,𝑛−𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌+𝐼−𝑠^∧_𝑌,𝑛+1𝜌_𝑌Tr𝜎^∧(𝑞)_𝑌,𝑛−𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛, (4.26) hence 2−𝐼−𝑠^∧_𝑌,𝑛+1𝜌_𝑌Tr𝜎^∧(𝑞)_𝑌,𝑛−𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛≤Tr𝜎^∧(𝑞)_𝑌,𝑛−𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌. (4.27) Since the operators 𝜌_𝑌 are trace class, inf_{𝜑∈ℋ𝑌;‖𝜑‖=1}⟨𝜑,𝜌_𝑌𝜑⟩=0. Thus, by assumption (4.22) and the self-adjointness of the operators 𝐼−𝑠^∧_𝑌,𝑛+1𝜌_𝑌, one obtains 𝐼−𝑠^∧_𝑌,𝑛+1𝜌_𝑌=sup_{𝜑∈ℋ𝑌𝜑=1}𝜑,𝐼−𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝜑=max1−𝑠^∧_𝑌,𝑛+1inf_{𝜑∈ℋ𝑌𝜑=1}𝜑,𝜌_𝑌𝜑,𝑠^∧_𝑌,𝑛+1sup_{𝜑∈ℋ𝑌𝜑=1}𝜑,𝜌_𝑌𝜑−1=1. (4.28)
The rest of the proof of (4.23) is by induction with respect to 𝑘≥2.
(1^∘) (𝑘=2) By Theorem 3.1 for 𝑛≥2 one has1(𝑛+1)²2𝑛+1_∧𝜎⁽²⁾_𝑌,𝑛+1=𝑛𝑛+1^∧𝜎⁽¹⁾_𝑌,𝑛∧(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌−1(𝑛+1)²𝑛2^∧𝜎⁽²⁾_𝑌,𝑛𝐼⊗𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ⁽²⁾_𝑌. (4.29) Assumption (4.22) gives Tr|𝜎^∧(2)_𝑌,𝑛(𝐼⊗(𝑠^∧_𝑌,𝑛+1𝜌_𝑌))𝐴_ℋ⁽²⁾_𝑌|≤𝑠^∧_𝑌,𝑛+1‖𝜌_𝑌‖Tr|𝜎^∧(2)_𝑌,𝑛|≤2, hence, by (4.29), Remark 4.4, and the assumption 𝜎^∧(2)_𝑌,𝑛≈𝜎^∧(2)_𝑌,𝑛+1, one obtains 𝜎^∧(2)_𝑌,𝑛+𝜎^∧(2)_𝑌,𝑛𝐼⊗𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ⁽²⁾_𝑌𝑛≈2𝑛+1^∧𝜎⁽¹⁾_𝑌,𝑛∧(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌. (4.30) Thus, in view of equivalence (4.16) from Theorem 4.8 and Lemma 4.6, one has 𝜎^∧(2)_𝑌,𝑛+𝜎^∧(2)_𝑌,𝑛𝐼⊗𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ⁽²⁾_𝑌𝑛≈2𝑛+1^∧𝜎⁽¹⁾_𝑌,𝑛∧𝜎^∧(1)_𝑌,𝑛𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌. (4.31) Furthermore, assumption (4.22) implies that the trace norms of the operators on the r.h.s of (4.31) are uniformly bounded. Therefore, according to Remark 4.4, 𝜎^∧(2)_𝑌,𝑛−2𝜎^∧(1)_𝑌,𝑛∧𝜎^∧(1)_𝑌,𝑛𝐼⊗𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ⁽²⁾_𝑌≈0. (4.32) The explicit form of 𝜎^∧(2)_𝑌,𝑛, 𝜎^∧(1)_𝑌,𝑛∧𝜎^∧(1)_𝑌,𝑛 given by Theorem 3.4 implies that 𝜎^∧(2)_𝑌,𝑛−2𝜎^∧(1)_𝑌,𝑛∧𝜎^∧(1)_𝑌,𝑛 and (𝐼⊗(𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌))𝐴_ℋ⁽²⁾_𝑌 commute, which proves the self-adjointness of the operator on the l.h.s of (4.32). Thus conditions (4.32), (4.27) for 𝑞=2, (4.28), and Lemma 4.5 yield relation (4.23) for 𝑘=2.
(2^∘) Assuming validity of equivalence (4.23) for 𝑘∈{2,…,𝑞−1}, where 𝑞∈ℕ, 𝑞>2, its validity will be proved for 𝑘=𝑞.
By Theorem 3.1 for 𝑛≥𝑞 one has1(𝑛+1)^𝑞𝑞𝑛+1_∧𝜎^(𝑞)_𝑌,𝑛+1=1(𝑛+1)^𝑞−1𝑛𝑞−1^∧𝜎^(𝑞−1)_𝑌,𝑛∧(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌−1(𝑛+1)^𝑞𝑛𝑞^∧𝜎^(𝑞)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌. (4.33) Assumption (4.22) implies Tr𝜎^∧(𝑞)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌≤𝑠^∧_𝑌,𝑛+1𝜌_𝑌Tr𝜎^∧(𝑞)_𝑌,𝑛≤2, (4.34) hence, in view of (4.33), Remark 4.4, and the assumption 𝜎^∧(𝑞)_𝑌,𝑛≈𝜎^∧(𝑞)_𝑌,𝑛+1, 𝜎^∧(𝑞)_𝑌,𝑛+𝜎^∧(𝑞)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌≈𝑞!(𝑛+1)^𝑞−1𝑛𝑞−1^∧𝜎^(𝑞−1)_𝑌,𝑛∧(𝑛+1)⁻¹𝑠^∧_𝑌,𝑛+1𝜌_𝑌. (4.35) Thus, by relation (4.16) from Theorem 4.8, Lemma 4.6, and Remark 4.4, one has 𝜎^∧(𝑞)_𝑌,𝑛+𝜎^∧(𝑞)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌≈𝑞!𝑞−1!^∧𝜎^(𝑞−1)_𝑌,𝑛∧𝜎^∧(1)_𝑌,𝑛𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌, (4.36) since the trace norms of the operators on the r.h.s. of (4.36) are uniformly bounded, by assumption (4.22). Furthermore, in view of Lemma 4.6 and the inductive hypothesis 𝜎^{∧(𝑞−1)}_𝑌,𝑛≈(𝑞−1)!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_{𝑌,𝑛𝑞−1}, condition (4.36) yields 𝜎^∧(𝑞)_𝑌,𝑛+𝜎^∧(𝑞)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌≈𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_{𝑌,𝑛𝑞−1}∧𝜎^∧(1)_𝑌,𝑛𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌, (4.37) hence 𝜎^∧(𝑞)_𝑌,𝑛−𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_{𝑌,𝑛𝑞}𝐼^{⊗(𝑞−1)}⊗𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌𝐴_ℋ^(𝑞)_𝑌≈0. (4.38) From the explicit form of 𝜎^∧(𝑞)_𝑌,𝑛, 𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛 given by Theorem 3.4 one finds that 𝜎^∧(𝑞)_𝑌,𝑛−𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛 and (𝐼^{⊗(𝑞−1)}⊗(𝐼+𝑠^∧_𝑌,𝑛+1𝜌_𝑌))𝐴_ℋ^(𝑞)_𝑌 commute, which proves the self-adjointness of the operator on the l.h.s of (4.38). Thus conditions (4.38), (4.27), (4.28), and Lemma 4.5 yield 𝜎^∧(𝑞)_𝑌,𝑛≈𝑞!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_{𝑌,𝑛𝑞}. Validity of relation (4.23) has been proved.
Now turn to equivalence (4.25). Similarly to (4.27) one has 2−𝐼+𝑠^∨_𝑌,𝑛+1𝜌_𝑌Tr𝜎^∨(𝑞)_𝑌,𝑛−𝑞!𝜎^∨(1)_𝑌,𝑛∨⋯∨𝜎^∨(1)_𝑌,𝑛≤Tr𝜎^∨(𝑞)_𝑌,𝑛−𝑞!𝜎^∨(1)_𝑌,𝑛∨⋯∨𝜎^∨(1)_𝑌,𝑛𝐼^{⊗(𝑞−1)}⊗𝐼−𝑠^∨_𝑌,𝑛+1𝜌_𝑌𝑆_ℋ^(𝑞)_𝑌. (4.39) Furthermore, according to assumption (4.24), 2−𝐼+𝑠^∨_𝑌,𝑛+1𝜌_𝑌≥2−1+𝑠^∨_𝑌,𝑛+1𝜌_𝑌≥1−𝜖>0. (4.40) The rest of the proof of (4.25) is by induction with respect to 𝑘≥2 and proceeds analogously to the proof of (4.23) with condition (4.28) replaced by (4.40) and the operators 𝐼∓𝑠^∧_𝑌,𝑛+1𝜌_𝑌 replaced by 𝐼±𝑠^∨_𝑌,𝑛+1𝜌_𝑌 (inversion of signs).

Theorem 4.9 allows to replace (𝑛,𝑘)-contractions of antisymmetric and symmetric product density operators by antisymmetric and symmetric products of 1-particle contractions, respectively, if the number 𝑛 of particles in the system is large. Further simplification, consisting in replacement of antisymmetric and symmetric products by tensor products, will be now proved possible. To this end weaker conditions on the asymptotic equivalence relation will be imposed.

Definition 4.10. Fix 𝑘∈ℕ and 𝑑∈ℝ, 𝑑>0. Families {𝐴_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ}, {𝐵_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} of operators 𝐴_𝑌,𝑛,𝐵_𝑌,𝑛∈𝒯(ℋ_𝑌^⊗𝑘) are called weakly asymptotically equivalent (symbolically: 𝐴_𝑌,𝑛∼𝐵_𝑌,𝑛), if 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr(𝐴_𝑌,𝑛−𝐵_𝑌,𝑛)𝐶_𝑌,𝑛=0 for every family {𝐶_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} of operators of the form 𝐶_𝑌,𝑛=⨂^𝑘_𝑖=1𝐶^(𝑖)_𝑌,𝑛, where 𝐶^(𝑖)_𝑌,𝑛∈ℬ(ℋ_𝑌) (𝑖∈{1,…,𝑘}, (𝑌,𝑛)∈ℳ(Ω)×ℕ) are operators with uniformly bounded operator norms.

The relation ~ has the properties analogous to the properties of the relation ≈ from Remark 4.4.

Definition 4.11. Let 𝑘∈ℕ, 𝑘≥2. Fix 𝜋∈𝑆_𝑘. A set 𝑋⊂{1,…,𝑘} is called a cyclic set of the permutation 𝜋, if 𝑋={𝑙₁,…,𝑙_𝑞} for some 𝑙₁,…,𝑙_𝑞∈{1,…,𝑘}, 𝑞∈{2,…,𝑘}, such that 𝜋(𝑙_𝑠)=𝑙_𝑠+1 for every 𝑠∈{1,…,𝑞−1}, and 𝜋(𝑙_𝑞)=𝑙₁. A singleton {𝑙}⊂{1,…,𝑘} such that 𝜋(𝑙)=𝑙 is also called a cyclic set of the permutation 𝜋.

Note that the set {1,…,𝑘} from the above definition can be represented as the union of disjoint cyclic sets of 𝜋.

Lemma 4.12. Let 𝑘∈ℕ, 𝑘≥2. If 𝐵⁽¹⁾,…,𝐵^(𝑘)∈𝒯(ℋ_𝑌) then 𝑘!Tr𝐵⁽¹⁾_𝑌,𝑛⊗⋯⊗𝐵^(𝑘)_𝑌,𝑛𝐴_ℋ^(𝑘)_𝑌=_{𝜋∈𝑆𝑘}sgn𝜋^𝑝(𝜋)_𝑗=1Tr^𝑞𝑗_𝑠=1𝐵^{(𝑙𝑗,𝑠)}_𝑌,𝑛, (4.41) where 𝑝(𝜋)∈{1,…,𝑘} is the number of disjoint cyclic sets of 𝜋, indexed by 𝑗, and 𝑞_𝑗 denotes the number of elements of the 𝑗th cyclic set of 𝜋, which is {𝑙_𝑗,1,…,𝑙_{𝑗,𝑞𝑗}}, where 𝜋𝑙_{𝑗,𝑞𝑗}=𝑙_𝑗,1,for𝑞_𝑗≥2,𝜋𝑙_𝑗,𝑠=𝑙_𝑗,𝑠+1,𝑠=1,…,𝑞_𝑗−1. (4.42) Clearly, ∑^𝑝(𝜋)_𝑗=1𝑞_𝑗=𝑘 and ⋃^𝑝(𝜋)_𝑗=1⋃^𝑞𝑗_𝑠=1{𝑙_𝑗,𝑠}={1,…,𝑘}.

Proof. Let {𝜑_𝑖}_𝑖∈ℕ be an orthonormal basis of ℋ_𝑌. One has 𝑘!Tr𝐵⁽¹⁾_𝑌,𝑛⊗⋯⊗𝐵^(𝑘)_𝑌,𝑛𝐴_ℋ^(𝑘)_𝑌=_{𝜋∈𝑆𝑘}sgn𝜋^𝑝(𝜋)_𝑗=1𝑀_𝑗, (4.43) where 𝑀_𝑗∶=^∞_{𝑖𝑙𝑗,1=1}…^∞_{𝑖𝑙𝑗𝑗,𝑞=1}𝜑_{𝑖𝑙𝑗,1},𝐵^(𝑙𝑗,1)𝜑_{𝑖)𝜋(𝑙𝑗,1}⋯𝜑_{𝑖𝑙𝑗𝑗,𝑞},𝐵^{(𝑙𝑗𝑗,𝑞)}𝜑_{𝑖𝑗)𝜋(𝑙𝑗,𝑞}. (4.44) If 𝑞_𝑗>2 for some 𝑗∈{1,…,𝑝(𝜋)} then, by condition (4.42) and Parseval's formula, 𝑀_𝑗=^∞_{𝑖𝑙𝑗,1=1}…^∞_{𝑖𝑙𝑗𝑗,𝑞=1}𝜑_{𝑖𝑙𝑗,1},𝐵^(𝑙𝑗,1)𝜑_{𝑖𝑙𝑗,2}𝜑_{𝑖𝑙𝑗,2},𝐵^(𝑙𝑗,2)𝜑_{𝑖𝑙𝑗,3}⋯𝜑_{𝑖𝑙𝑗𝑗,𝑞},𝐵^{(𝑙𝑗𝑗,𝑞)}𝜑_{𝑖𝑙𝑗,1}=^∞_{𝑖𝑙𝑗,1}^∞_{=1𝑖𝑙𝑗,3=1}…^∞_{𝑖𝑙𝑗𝑗,𝑞=1}𝜑_{𝑖𝑙𝑗,1},𝐵^(𝑙𝑗,1)𝐵^(𝑙𝑗,2)𝜑_{𝑖𝑙𝑗,3}⋯𝜑_{𝑖𝑙𝑗𝑗,𝑞},𝐵^{(𝑙𝑗𝑗,𝑞)}𝜑_{𝑖𝑙𝑗,1}. (4.45) Performing successive summations one then obtains 𝑀_𝑗=^∞_{𝑖𝑙𝑗,1=1}𝜑_{𝑖𝑙𝑗,1},^𝑞𝑗_𝑠=1𝐵^{(𝑙𝑗,𝑠)}𝜑_{𝑖𝑙𝑗,1}=Tr^𝑞𝑗_𝑠=1𝐵^{(𝑙𝑗,𝑠)}. (4.46) The derivation of the above formula for 𝑞_𝑗=1,2, after simplifications, proceeds analogously. This completes the proof of (4.41), in view of (4.43).

Lemma 4.13. One has 𝑑−lim𝜎_{𝑛,𝜇(𝑌)→∞}^∧(1)_𝑌,𝑛=0, (4.47) and if 𝜎^∨(2)_𝑌,𝑛≈2𝜎^∨(1)_𝑌,𝑛∨𝜎^∨(1)_𝑌,𝑛 (see Theorem 4.9) then 𝑑−lim𝜎_{𝑛,𝜇(𝑌)→∞}^∨(1)_𝑌,𝑛=0. (4.48)

Proof. To prove (4.47) it suffices to observe that, according to Theorem 2.3, 𝜎^∧(1)_𝑌,𝑛=L¹_𝑛1𝜉^∧_𝑌,𝑛𝜌_𝑌^∧𝑛≤1𝑛1𝜉^∧_𝑌,𝑛𝜌_𝑌^∧𝑛≤1𝑛1𝜉^∧_𝑌,𝑛Tr𝜌_𝑌^∧𝑛=1𝑛. (4.49)
Now (4.48) will be proved. Let {𝜑_𝑖}_𝑖∈ℕ be an orthonormal basis of ℋ_𝑌 for fixed 𝑌∈ℳ(Ω). ThenTr2𝜎^∨(1)_𝑌,𝑛∨𝜎^∨(1)_𝑌,𝑛=2Tr𝜎^∨(1)_𝑌,𝑛⊗𝜎^∨(1)_𝑌,𝑛𝑆_ℋ⁽²⁾_𝑌=_𝜋∈𝑆2^∞_{𝑖1,𝑖2=1}𝜑_𝑖1,𝜎^∨(1)_𝑌,𝑛𝜑_𝑖𝜋(1)𝜑_𝑖2,𝜎^∨(1)_𝑌,𝑛𝜑_𝑖𝜋(2)=Tr𝜎^∨(1)_𝑌,𝑛²+Tr𝜎^∨∨(1)_𝑌,𝑛𝜎⁽¹⁾_𝑌,𝑛. (4.50) Taking into account (4.50), the relation 𝜎^∨(2)_𝑌,𝑛≈2𝜎^∨(1)_𝑌,𝑛∨𝜎^∨(1)_𝑌,𝑛, Definition 4.2 for 𝐶_𝑌,𝑛=𝐼^⊗2, and the equality Tr𝜎^∨(2)_𝑌,𝑛=Tr𝜎^∨(1)_𝑌,𝑛=1, one obtains 𝑑−lim_{𝑛,𝜇(𝑌)→∞}Tr𝜎^∨∨(1)_𝑌,𝑛𝜎⁽¹⁾_𝑌,𝑛=0. (4.51) Furthermore, 𝜎^∨(1)_𝑌,𝑛𝜑²=𝜑,𝜎^∨∨(1)_𝑌,𝑛𝜎⁽¹⁾_𝑌,𝑛𝜑≤Tr𝜎^∨∨(1)_𝑌,𝑛𝜎⁽¹⁾_𝑌,𝑛 (4.52) for every 𝜑∈ℋ_𝑌 such that ‖𝜑‖=1, hence (4.51) yields (4.48).

Notice that (4.47) can be also proved analogously to (4.48) under the additional assumption 𝜎^∧(2)_𝑌,𝑛≈2𝜎^∧(1)_𝑌,𝑛∧𝜎^∧(1)_𝑌,𝑛.

The proof of the next theorem for 𝑘=2 was given in [11, 12].

Theorem 4.14 (Asymptotic formulae II). Let 𝑘∈ℕ, 𝑘≥2. One has 𝑘!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_{𝑌,𝑛𝑘}∼𝜎^∧(1)_𝑌,𝑛⊗⋯⊗𝜎^∧(1)_{𝑌,𝑛𝑘}, (4.53) and if 𝑑−lim_{𝑛,𝜇(𝑌)→∞}‖𝜎^∨(1)_𝑌,𝑛‖=0 (see Lemma 4.13) then 𝑘!𝜎^∨(1)_𝑌,𝑛∨⋯∨𝜎^∨(1)_{𝑌,𝑛𝑘}∼𝜎^∨(1)_𝑌,𝑛⊗⋯⊗𝜎^∨(1)_{𝑌,𝑛𝑘}. (4.54)

Proof. First (4.53) will be proved. Fix a family {𝐶_𝑌,𝑛}_{(𝑌,𝑛)∈ℳ(Ω)×ℕ} of operators such as in Definition 4.10 and set 𝐵^(𝑟)_𝑌,𝑛∶=𝜎^∧(1)_𝑌,𝑛𝐶^(𝑟)_𝑌,𝑛,𝑟=1,…,𝑘. (4.55) Then, by Lemma 4.12, one has Tr𝑘!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛𝐶⁽¹⁾_𝑌,𝑛⊗⋯⊗𝐶^(𝑘)_𝑌,𝑛=𝑘!Tr𝐵⁽¹⁾_𝑌,𝑛⊗⋯⊗𝐵^(𝑘)_𝑌,𝑛𝐴_ℋ^(𝑘)_𝑌=_{𝜋∈𝑆𝑘}sgn𝜋^𝑝(𝜋)_𝑗=1Tr^𝑞𝑗_𝑠=1𝐵^{(𝑙𝑗,𝑠)}_𝑌,𝑛=Tr𝜎^∧(1)_𝑌,𝑛⊗⋯⊗𝜎^∧(1)_𝑌,𝑛𝐶⁽¹⁾_𝑌,𝑛⊗⋯⊗𝐶^(𝑘)_𝑌,𝑛+_{𝜋∈𝑆𝑘𝜋≠Id}sgn𝜋^𝑝(𝜋)_𝑗=1Tr^𝑞𝑗_𝑠=1𝐵^{(𝑙𝑗,𝑠)}_𝑌,𝑛. (4.56) Thus Tr𝑘!𝜎^∧(1)_𝑌,𝑛∧⋯∧𝜎^∧(1)_𝑌,𝑛−𝜎^∧(1)_𝑌,𝑛⊗⋯⊗𝜎^∧(1)_𝑌,𝑛𝐶⁽¹⁾_𝑌,𝑛⊗⋯⊗𝐶^(𝑘)_𝑌,𝑛=_{𝜋∈𝑆𝑘𝜋≠Id}sgn𝜋^𝑝(𝜋)_𝑗=1Tr^𝑞𝑗_𝑠=1𝐵^{(𝑙𝑗,𝑠)}_𝑌,𝑛. (4.57) Now, let 𝜋∈𝑆_𝑘, 𝜋≠Id, be fixed. If 𝑞_𝑗=1 for some 𝑗∈{1,…,𝑝(𝜋)} then Tr^𝑞𝑗_𝑠=1𝐵^{(𝑙𝑗,𝑠)}_𝑌,𝑛≡Tr𝐵^(𝑙𝑗,1)_𝑌,𝑛≤𝐶^(𝑙𝑗,1)_𝑌,𝑛Tr𝜎^∧(1)_𝑌,𝑛=𝐶^(𝑙𝑗,1)_𝑌,𝑛, (4.58) whereas if 𝑞_𝑗≥2 then Tr^𝑞𝑗_𝑠=1𝐵^{(𝑙𝑗,𝑠)}_𝑌,𝑛≤^𝑞𝑗−1_𝑠=1𝐵^{(𝑙𝑗,𝑠)}_𝑌,𝑛Tr𝐵^{(𝑙𝑗𝑗,𝑞)}_𝑌,𝑛≤𝜎^∧(1)_𝑌,𝑛^{𝑞𝑗−1𝑞𝑗−1}_𝑠=1𝐶^{(𝑙𝑗,𝑠)}_𝑌,𝑛𝐶^{(𝑙𝑗𝑗,𝑞)}_𝑌,𝑛Tr𝜎^∧(1)_𝑌,𝑛≤𝜎^∧(1)_𝑌,𝑛^{𝑞𝑗𝑞−1𝑗}_𝑠=1𝐶^{(𝑙𝑗,𝑠)}_𝑌,𝑛. (4.59) Since 𝜋≠Id, there exists at least one 𝑗∈{1,…,𝑝(𝜋)} such that 𝑞_𝑗≥2, hence ^𝑝(𝜋)_𝑗=1Tr^𝑞𝑗_𝑠=1𝐵^{(𝑙𝑗,𝑠)}_𝑌,𝑛≤𝜎^𝑝(𝜋)_𝑗=1^∧(1)_𝑌,𝑛^{𝑞𝑗−1𝑝(𝜋)𝑞}_𝑗=1^𝑗_𝑠=1𝐶^{(𝑙𝑗,𝑠)}_𝑌,𝑛=𝐶𝜎_𝑌,𝑛^𝑝(𝜋)_𝑗=1^∧(1)_𝑌,𝑛^𝑞𝑗−1 (4.60) and at least one exponent 𝑞_𝑗−1 is nonzero. Thus, by the uniform boundedness of the norms ‖𝐶_𝑌,𝑛‖ and Lemma 4.13, the termodynamic limit of the l.h.s of (4.57) equals 0, which proves the validity of relation (4.53).
The proof of relation (4.54), after discarding the permutation signs and replacing ∧ by ∨, proceeds analogously.

Appendix

Product Integral Kernels of Trace Class Operators

In this section theorems concerning product integral kernels, exploited in Section 3, are formulated.

Fix the Hilbert space ℋ_𝑌∶=𝐿²(𝑌,𝜇) over the field 𝕂=ℂ or ℝ, where the measure 𝜇is separable and 𝜎-finite. For every 𝑛∈ℕ the space ℋ_𝑌^⊗𝑛 is identified with 𝐿²(𝑌^𝑛,𝜇^⊗𝑛). Unless otherwise stated, elements of 𝐿² spaces are identified with their representatives and denoted by the same symbols.

Let 𝒦∈𝐿²(𝑌²,𝜇^⊗2). In the case of the integral operator 𝐾∶𝐻_𝑌→𝐻_𝑌 defined for every 𝜑∈𝐻_𝑌 and𝜇-a.a. 𝑥∈𝑌 by𝐾𝜑(𝑥)=_𝑌𝒦𝑥,𝑦𝜑𝑦d𝜇𝑦 (A.1) both 𝒦 regarded as an element of 𝐿²(𝑌²,𝜇^⊗2) as well as its arbitrary representative is called an integral kernel of 𝐾. The kernel 𝒦 is unique as an element of 𝐿²(𝑌²,𝜇^⊗2) but a representative of 𝒦 of a special form, given in Lemma A.3 and Definition A.4, is useful in computations of the trace of 𝐾.

Let ℋ𝒮(ℋ_𝑌) be the space of Hilbert-Schmidt operators on ℋ_𝑌 with the inner product defined by ⟨𝐴,𝐵⟩_{ℋ𝒮(ℋ𝑌)}∶=Tr𝐴^∗𝐵 and the induced norm denoted by ‖⋅‖_{ℋ𝒮(ℋ𝑌)}. In the sequel use is made of the following theorem, the proof of which can be found in [19].

Theorem A.1. An operator 𝐾∈ℬ(𝐻_𝑌) is Hilbert-Schmidt if and only if it is an integral operator with an integral kernel 𝒦∈𝐿²(𝑌²,𝜇^⊗2). Furthermore, ‖𝐾‖_{ℋ𝒮(𝐻𝑌)}=‖𝒦‖_𝐿²_(𝑌²_,𝜇⊗2).

Corollary A.2. Let 𝐾,𝐺∈ℋ𝒮(𝐻_𝑌) and let 𝒦,𝒢∈𝐿²(𝑌²,𝜇^⊗2) be integral kernels of the operators 𝐾, 𝐺, respectively. Then ⟨𝐾,𝐺⟩_{ℋ𝒮(𝐻𝑌)}=⟨𝒦,𝒢⟩_𝐿²_(𝑌²_,𝜇⊗2).

Recall that 𝐾∈ℬ(𝐻_𝑌) is a trace class operator if and only if there exist operators 𝐾₁,𝐾₂∈ℋ𝒮(𝐻_𝑌) such that 𝐾=𝐾₁𝐾₂. Moreover, Tr𝐾=⟨𝐾^∗₁,𝐾₂⟩_{ℋ𝒮(𝐻𝑌)}. This fact, Theorem A.1, and Corollary A.2 imply the following lemma, in which elements of the 𝐿² space are distinguished from their representatives. The element of the 𝐿² space represented by a function 𝑓 is denoted by [𝑓].

Lemma A.3. Let 𝐾∈𝒯(𝐻_𝑌), 𝐾=𝐾₁𝐾₂, where 𝐾₁,𝐾₂∈ℋ𝒮(𝐻_𝑌). Let [𝒦₁],[𝒦₂]∈𝐿²(𝑌²,𝜇^⊗2) be integral kernels of 𝐾₁,𝐾₂. Then for any choice of representatives 𝒦₁∈[𝒦₁], 𝒦₂∈[𝒦₂] the function 𝒦∶𝑌×𝑌→𝕂 defined for 𝜇^⊗2-a.a. (𝑥,𝑦)∈𝑌×𝑌 by 𝒦𝑥,𝑦=_𝑌𝒦₁(𝑥,𝑧)𝒦₂𝑧,𝑦d𝜇(𝑧) (A.2) is 𝜇^⊗2-square integrable and it is an integral kernel of 𝐾. The function ℒ∶𝑌→𝕂 defined for μ-a.a. 𝑥∈𝑌 by ℒ(𝑥)=𝒦(𝑥,𝑥) is μ-integrable. Moreover, Tr𝐾=_𝑌ℒ(𝑥)d𝜇(𝑥)≡_𝑌𝒦(𝑥,𝑥)d𝜇(𝑥). (A.3)

Definition A.4. Under the assumptions of Lemma A.3, the function 𝒦 given by formula (A.2) (for any choice of representatives 𝒦₁,𝒦₂ of [𝒦₁],[𝒦₂]) is called a product integral kernel of 𝐾.

Notice that for 𝜇being the Lebesgue measure on [0,1]×[0,1] formula (A.3) is valid, for example, if 𝒦 is any continuous function.

In the following lemma, which follows from Lemma A.3, the function 𝒦₀ need not be a product integral kernel of 𝐾₀ but the integral formula for the trace of 𝐾₀ still holds for 𝒦₀.

Lemma A.5. Let 𝑘,𝑛∈ℕ, 𝑘<𝑛, and let 𝒦 be a product integral kernel of 𝐾∈𝒯(𝐻_𝑌^⊗𝑛)≡𝒯(𝐿²(𝑌^𝑛,𝜇^⊗𝑛)). Then the function 𝒦₀∶𝑌^𝑘×𝑌^𝑘→𝕂 defined for 𝜇^⊗2𝑘-a.a. (𝑥^′,𝑦^′)∈𝑌^𝑘×𝑌^𝑘 by 𝒦₀𝑥^′,𝑦^′=_{𝑌𝑛−𝑘}𝒦𝑥^′,𝑥^′′,𝑦^′,𝑥^′′d𝜇^{⊗(𝑛−𝑘)}𝑥^′′ (A.4) is 𝜇^⊗2𝑘-square integrable and the integral operator 𝐾₀ with the kernel 𝒦₀ belongs to 𝒯(𝐻_𝑌^⊗𝑘). For every 𝜒,𝜑∈𝐻_𝑌^⊗𝑘 and every orthonormal basis {𝜓_𝑖}_𝑖∈ℕ of 𝐻_𝑌^{⊗(𝑛−𝑘)} one has 𝜒,𝐾₀𝜑_{𝐻𝑌⊗𝑘}=^∞_𝑖=1𝜒⊗𝜓_𝑖,𝐾(𝜑⊗𝜓_𝑖)_{𝐻𝑌⊗𝑛}. (A.5) The function ℒ₀∶𝑌^𝑘→𝕂 defined for 𝜇^⊗𝑘-a.a. 𝑥^′∈𝑌^𝑘 by ℒ₀(𝑥^′)=𝒦₀(𝑥^′,𝑥^′) is 𝜇^⊗𝑘-integrable. Moreover, _𝑌𝑘𝒦₀𝑥^′,𝑥^′d𝜇^⊗𝑘𝑥^′≡_𝑌𝑘ℒ₀𝑥^′d𝜇^⊗𝑘𝑥^′=Tr𝐾₀=Tr𝐾. (A.6)

Corollary A.6. Under the assumptions of Lemma A.5, if 𝐶∈ℬ(𝐻_𝑌^⊗𝑘) then Tr𝐶𝐾₀=Tr(𝐶⊗𝐼^{⊗(𝑛−𝑘)})𝐾.

Acknowledgments

This paper presents the results of a part of the author's MS thesis [2] written in the Institute of Physics, Nicolaus Copernicus University, Toruń, under the supervision of Professor Jan Maćkowiak. The author wishes to express his gratitude to Professor Maćkowiak for helpful suggestions and remarks. Professor Maćkowiak prepared also, on his own initiative, the English translation of appropriate parts of the author's thesis, which was useful for the author in editing of the present paper.