A Special Class of Infinite Dimensional Dirac Operators on the Abstract Boson-Fermion Fock Space

Spectral properties of a special class of infinite dimensional Dirac operators Q(α) on the abstract boson-fermion Fock space F(H,K) associated with the pair (H,K) of complex Hilbert spaces are investigated, where α ∈ C is a perturbation parameter (a coupling constant in the context of physics) and the unperturbed operator Q(0) is taken to be a free infinite dimensional Dirac operator. A variety of the kernel of Q(α) is shown. It is proved that there are cases where, for all sufficiently large |α| with α < 0, Q(α) has infinitely many nonzero eigenvalues even if Q(0) has no nonzero eigenvalues. Also Fredholm property of Q(α) restricted to a subspace ofF(H,K) is discussed.


Introduction
In a previous paper [1] (cf. [2]), the author introduced a general class of infinite dimensional Dirac type operators   on the abstract boson-fermion Fock space F(H, K) associated with the pair (H, K) of complex Hilbert spaces (for the definition of F(H, K) and   , see Section 2), where  is a densely defined closed linear operator from H to K. The operator   gives an infinite dimensional and abstract version of finite dimensional Dirac type operators.In applications to physics,   unifies self-adjoint supercharges (generators of supersymmetry) of some supersymmetric quantum field models (e.g., [3][4][5][6]).
In the paper [1], basic properties of   are discussed.As for the spectral properties of   , only the zero-eigenvalue of   is considered.Moreover, a class of perturbations  for   is introduced and, under a suitable condition, a path (functional) integral representation for the index of the perturbed operator   () :=   +  restricted to a subspace of F(H, K) is established.The (essential) self-adjointness of   () is partially discussed in [7].Analysis of other properties of   () including spectral ones except for the zero-eigenvalue has been left open.
In this paper, we undertake a comprehensive operatortheoretical analysis for   ().But, as a preliminary, we consider, in the present paper, only the case where  is a simple form and see what kind of phenomena occurs under the perturbation .
The outline of the present paper is as follows.In Section 2 we review briefly some contents in [1].In Section 3 we identify the spectra of   .This is done via spectral analysis of a second quantization operator on the boson-fermion Fock space F(H, K).In Section 4, we introduce a perturbation  =  ,V () with  ∈ H \ {0} and V ∈ K \ {0}, where  ∈ C is a perturbation parameter (a coupling constant in the context of physics).The perturbed operator () :=   +  ,V () is the main object of our analysis in the present paper.We prove the self-adjointness of () and the essential selfadjointness of it on a suitable dense subspace of F(H, K) with some other properties (Theorem 14).Also the spectra of () are identified (Theorem 16).Moreover, it is shown that the domain of () is equal to that of (0) =   for all sufficiently small || (Theorem 17).Section 5 is devoted to analysis of the kernel of ().We see that the kernel property of () may be sensitive to conditions for (,  * , ).In Section 6 we consider Fredholm property of () + , the restriction of () to a subspace of F(H, K), in comparison with that of (  ) + = (0) + .We obtain some classification on Fredholm property of () + (Theorem 26).An interesting phenomenon occurs in the following sense: for a constant  0 ̸ = 0 (resp.,  0 ̸ = 0), ( 0 ) + is not semi-Fredholm even if (  ) + is Fredholm (resp., semi-Fredholm).In the last For a complex Hilbert space H and a nonnegative integer , we denote by ⊗   H (resp., ∧  (H)) the -fold symmetric (resp., antisymmetric) tensor product Hilbert space of H with convention ⊗ 0  H := C (the complex number field) and ∧ 0 (H) := C. In terms of these Hilbert spaces, one can construct two types of larger Hilbert spaces.The one is the boson (symmetric) Fock space over H, denoted by F  (H), which is defined to be the complete infinite direct sum of ⊗   H,  ≥ 0: where, for a complex Hilbert space X, we denote its inner product (resp., norm) by ⟨⋅, ⋅⟩ (linear in the second variable) (resp., ‖ ⋅ ‖).The other is the fermion (antisymmetric) Fock space over H, denoted by F  (H), which is defined to be the complete infinite direct sum of ∧  (H),  ≥ 0: ( For general theories of boson and fermion Fock spaces, see, for example, [8][9][10]. Let K be a complex Hilbert space.Then the abstract boson-fermion Fock space associated with the pair (H, K) is defined as the tensor product Hilbert space of F  (H) and F  (K).
For each  ∈ H, we denote by () the bosonic annihilation operator on F  (H) with test vector ; that is, () is the densely defined closed linear operator acting in F  (H) such that its adjoint () * is given by the following form: where   is the symmetrization operator acting on the -fold tensor product Hilbert space ⊗  H. Let called the bosonic Fock vacuum.Then, for all  ∈ H, For each  ∈ K, there exists a unique linear operator () on F  (K), called the fermionic annihilation operator with test vector , which has the following properties: (i) (()) = F  (K) and () is bounded with operator norm ‖()‖ = ‖‖; (ii) its adjoint () * is of the following form: where   is the antisymmetrization operator acting on the -fold tensor product Hilbert space ⊗  H.The following canonical anticommutation relations (CAR) hold: where {, } :=  + .In particular, one has The vector is called the fermionic Fock vacuum and obeys 2.2.Infinite Dimensional Exterior Differential Operators.For a subset D of a Hilbert space, we denote by L(D) the subspace algebraically spanned by all the vectors of D.
Let  be a densely defined closed linear operator from H to K and define where  ∞ () := ∩ ∞ =1 (  ) for a linear operator  on a Hilbert space (the symbol  1 ⋅ ⋅ ⋅   Ω with  = 0 should read Ω).It follows that D ∞  is dense in F(H, K).The following proposition is proved in [1].Proposition 1.There exists a unique densely defined closed linear operator   on F(H, K) such that the following (i) and (ii) hold: for all vectors Ψ of the form (13) with  ≥ 1.In the case  = 0, we have  *  Ψ = 0.In particular,  *  leaves D ∞  invariant.
(iv) ( 2  ) = (  ) and, for all Ψ ∈ (  ),  2  Ψ = 0. (v) Let  be a bounded linear operator from H to K with () = H.Then, for all Ψ ∈ D ∞  and ,  ∈ C, The operator   is an abstract version of finite dimensional exterior differential operators, including an infinite dimensional version of them if H and K are infinite dimensional.
For a self-adjoint operator  on H, we define an operator Γ ()   () on ⊗   H by where  denotes identity and, for a closable linear operator ,  denotes the closure of .For each  ≥ 0, Γ ()  () is self-adjoint.The second quantization Γ  () of  on F  (H) is defined by which is self-adjoint (see, e.g., [9, Section 5.2], [11, p.302, Example 2]).It follows that Similarly, for a self-adjoint operator  on K, we can define the second quantization Γ  () of  on F  (K) by where Journal of Mathematics acting in ∧  (K).Similarly to the case of Γ  (), one has The number operator on F  (K) is defined by Then the operator giving the orthogonal decomposition Let  + and  − be the orthogonal projections onto F + (H, K) and F − (H, K), respectively.Then For notational simplicity we set We introduce an operator acting in F(H, K), which is nonnegative and self-adjoint.It follows from ( 21) and (24) that Theorem 2 (see [1]).(i) The operator   is self-adjoint and essentially self-adjoint on every core of   ().In particular,   is essentially self-adjoint on D ∞  .(ii) The operator Γ leaves (  ) invariant and The following operator equalities hold: Remark 3. The quadruple {F(H, K),   ,   (), Γ} is an abstract form of models of free supersymmetric quantum field theory, where   ,   (), and Γ are a self-adjoint supercharge, a supersymmetric Hamiltonian, and a sign operator of states (a "fermion number operator"), respectively [1].
For a linear operator , we set the nullity of .

Fredholm Property and Index.
Let  be a self-adjoint operator on F(H, K) such that Γ leaves () invariant satisfying Γ + Γ = 0 on ().This class of operators  may be an abstract category of Dirac operators on the abstract boson-fermion Fock space F(H, K).
Then it is easy to show that the operator  + defined by is a densely defined closed linear operator from provided that at least one of nul  + and nul  * + is finite.Note that the index of  + (e.g., [12, Chapter IV, Section 5]).
As for the Fredholm property of   , one has the following theorem.

Spectra of 𝐻 𝐵𝐹 (𝐴) and 𝑄 𝐴
Now we go into analyzing new aspects in the operator theory of   and its perturbations.In the previous paper [1], the identification of the spectra of   () and   was not discussed except for zero-eigenvalues.This section is devoted to identifying completely the spectra of   () and   .
In what follows, we assume that H and K are separable.We say that a subset  of R is symmetric with respect to the origin if  ∈  implies that − ∈ .
The main results in this section are as follows.Theorem 6.
(43) Theorem 7. The spectrum (  ) and the point spectrum   (  ) of   are symmetric with respect to the origin and To prove these theorems, we need a series of lemmas.
Lemma 8. Let  be a self-adjoint operator on a Hilbert space.
A sufficient condition for the spectra of a self-adjoint operator to be symmetric with respect to the origin is given in the following lemma.

Lemma 9.
Let  be a self-adjoint operator on a Hilbert space X.Suppose that there exists a bounded self-adjoint operator  on X such that  2 =  and  leaves () invariant satisfying  +  = 0   () . (48) Then () and   () are symmetric with respect to the origin and, for all  ∈   (), Proof.Note that  is unitary with  −1 = .Equation (48) implies the operator equality  = (−) −1 .Hence  is unitarily equivalent to −.Thus the desired results follow.
For a self-adjoint operator  on a Hilbert space K and  = 1, 2, . .., we define where, for a finite set , # denotes the number of the elements of  and   () denotes the discrete spectrum of  (the set of isolated eigenvalues of  with finite multiplicity).
Proof of Theorem 7. By Theorem 2(ii) and Lemma 9, (  ) and   (  ) are symmetric with respect to the origin and (45) holds.By Lemma 8 and Theorem 2(iii), we have By these facts and Theorem 6, we obtain (44).

A Class of Perturbed Infinite Dimensional Dirac Operators
In this section we consider a class of perturbations of   .It would be natural to perturb   through a perturbation of   .

Self-Adjointness of 𝑄(𝛼).
We define a linear operator  ,V from H to K by It is obvious that  ,V is a bounded operator (a one-rank operator).Hence is a densely defined closed linear operator with (()) = ().
The following lemma is a key fact.

Spectra of 𝑄(𝛼).
We have (77).Hence, applying Theorem 7 with  replaced by (), we obtain the following result on the spectra of ().
Theorem 16.For all  ∈ C, (()) and   (()) are symmetric with respect to the origin and This theorem shows that the spectrum and the point spectrum of () are completely determined from those of () * () \ {0}.

Identification of the Domain of 𝑄(𝛼).
The following theorem gives a sufficient condition for the domain of () to be equal to that of the unperturbed operator   .

Kernel of 𝑄(𝛼)
In this section we investigate the kernel of ().For this purpose, we need some conditions on {, , V}.
Using Lemma 22, one can prove (ii)-(iv) in the same manner as in the proof of (i).