Spectral Properties of a Certain Class of Carleman Operators

The object of the present work is to construct all the generalized spectral functions of a certain class of Carleman operators in the Hilbert space L (X,μ) and establish the corresponding expansion theorems, when the deficiency indices are (1,1). This is done by constructing the generalized resolvents of A and then using the Stieltjes inversion formula. 1. Preliminaries The set of generalized resolvents of a symmetric operator A with defect indices (1, 1) was first derived independently by Naimark [15] and Krein [10]. The case of defect indices (m,m), m ∈ N is due to Krein [11]. Saakjan [19] extended Krein’s formula to the general case of defect indices (m,m), m ∈ N ∪ {∞}. In another form, the generalized resolvent formula for symmetric operators (including the case of non-densely defined operators) has been obtained by Straus [20, 21]. Let H be a Hilbert space endowed with the inner product (·, ·), and let A : D(A) ⊂ H −→ H be a densely defined closed linear operator whose range is denoted R(A). 1.1. Basic Spectral Properties. We say that λ ∈ C is a regular point of the operator A if the resolvent Rλ = (A− λI) −1 exists and is a bounded operator defined everywhere in H (I denotes the identity operator in H). In this case we say that λ belongs to ρ(A), the resolvent set of A. Rλ is an analytic operator function of λ on ρ(A). The number λ ∈ C is said to be an eigenvalue of A if there exists an f ∈ D(A) for which f 6= 0 and Af = λf . In this case, the operator A− λI is not injective, i.e., ker (A − λI) 6= {0}. The complement of ρ(A), in the complex plane, is denoted by σ(A) and is called the spectrum of A. A resolution of the identity [1] is a one-parameter family {Et}, −∞ < t < ∞, of orthogonal projection operators acting on a Hilbert space H , such that i) Es ≤ Et if s ≤ t (monotonicity); 2000 Mathematics Subject Classification : Primary 05C38, 15A15; Secondary 05A15, 15A18.