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

Abstract

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