SPECTRAL PROPERTIES OF A CERTAIN CLASS OF CARLEMAN OPERATORS

Abstract. 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 L2 (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)