On Convex Hull of Orthogonal Scalar Spectral Functions of a Carleman Operator

Let H be a (separable) Hilbert space and let A be a symmetric operator in H with defect indices (1, 1) . It is well known that the set W (A) of all generalized spectral functions of A is both convex and closed (in some natural topology). Consider the following problem: Describe a convex hull W0 (A) of orthogonal spectral functions of A. This problem has been solved by I.M. Glazman [7] for jacobi matrices corresponding to Hamburger moment problem. To explain his result let us recall that according to the Krein-Naimark formula for generalized resolvents of A the set W (A) is described as follows: Et ∈W (A) if and only if ∫ d (Etf, f) t− λ = D0 (λ)φ (λ) +D1 (λ) C0 (λ)φ (λ) + C1 (λ) , φ ∈ N, (1)


Introduction
Let H be a (separable) Hilbert space and let A be a symmetric operator in H with defect indices (1,1) .It is well known that the set W (A) of all generalized spectral functions of A is both convex and closed (in some natural topology).Consider the following problem: Describe a convex hull W 0 (A) of orthogonal spectral functions of A. This problem has been solved by I.M. Glazman [7] for jacobi matrices corresponding to Hamburger moment problem.To explain his result let us recall that according to the Krein-Naimark formula for generalized resolvents of A the set W (A) is described as follows: E t ∈ W (A) if and only if where D i ,C i (i = 1, 2) are entries of the resolvent matrix of A and f is a "scale" vector and N stands for the Nevanlinna class of functions holomorphic in C + with non-negative imaginary parts [9].Glazman [7] proved that E t ∈ W 0 (A) if and only if ϕ in (1) admits a representation ϕ (λ) = ϕ * (C 1 (λ) /C 0 (λ)) , ϕ * ∈ N. ( Though Glazman proved this result for Jacobi matrices, it remains valid for any symmetric operator A with defect indices (1,1) .This result may be found in S. M. Bahri [2,5,14].However, the orthogonal resolvents and the orthogonal spectral functions, which are those of the selfadjoint extensions in the same space, play a central role.We start by describing the closed convex hull of orthogonal resolvents of an abstract symmetric operator of defect indices (1, 1), then we study the convex hull of orthogonal spectral functions of a Carleman operator in the Hilbert space L 2 (X, µ) .Let us notice that the same problem for the differential operator of second order on (0, ∞) was considered in [8].

Resolvents of a symmetric operator of defect indices (1.1)
We begin by introducing the following sets: Φ the set of the analytical functions ϕ (z) on the unit disc M the set of all functions ϕ (z) ∈ Φ admitting the representation where S (t) is a monotonic nondecreasing function with total variation equal to one, i.e. 2π 0 dS (t) = 1; M 0 the set of all functions ϕ (z) ∈ M with S (t) a step function with a finite number of jumps.
1. M is closed under pointwise convergence and M 0 is dense in M.

M = Φ.
Proof: 1. We assume that for all z ∈ K According to a theorem by Helly [10], there exists a nondecreasing function S (t) with total variation equal to 1 and a subsequence n k such that lim in each point of continuity of S (t).Thus we have lim and the ensity of M 0 in M is then straightforward.
2. Now, we introduce two other sets : Φ + the set of all functions f (λ) analytic in the upper half-plane Π + = {λ ∈ C : λ > 0} with positive imaginary part, i.e. f (λ) ≥ 0, and M + the set of functions τ (λ) ∈ Φ + having the following representation ) being a nondecreasing function with total variation equal to to one.
It is easy to see that the homographic transformation establishes a bijection between the sets Φ and Φ We can also see that (4) establishes the same bijection enters M and M + .Indeed, we have for τ (λ) And making in the formula (5) the following change of variables we will have where But we know [8] that M + is dense in Φ + , consequently M is dense in Φ for pointwise convergence.As M is closed for this convergence then M =Φ.

2
Let A be a closed symmetric operator defined in a Hilbert space H with the scalar product (., .) .Its domain of definition D (A) is assumed to be dense in H.
where I is the identity operator in H.We recall that D (A) and the defect subspaces N λ and N λ are linearly independent.
In the sequel we suppose the operator A with defect indices (1, 1) , i.e. dim We now state the formula of generalized resolvents R ω (λ) of the operator A obtained in [2] and [4] is the resolvent of the selfadjoint extension [9] and The generalized resolvent defined by ( 6) is a resolvent of a selfadjoint extension of A if and only if ω (λ) = κ = const, |κ| = 1.We will call it orthogonal ( or canonical ) resolvent.
Thus we have for an we obtain Let us denote by R = R 0 the closed convex hull of orthogonal resolvents in strong topology.
for all λ, λ > 0. Finally, according to the lemma1 Let E κj t , j = 1, 2, ..., n be the orthogonal spectral functions connected with the orthogonal resolvents R κJ (λ) by (7).We denote by E 0 the convex hull of orthogonal spectral functions  and E = E 0 for strong topology.It is obvious that each E ω t ∈ E defines by ( 7) a generalized resolvent R ω (λ) ∈ R and, reciprocally, each generalized resolvent R ω (λ) ∈ R defines by ( 7) a generalized spectral function E ω t ∈ E.

Orthogonal scalar spectral functions of a Carleman operator
Let X be an arbitrary set, µ a σ-finite measure on X, L 2 (X, µ) the Hilbert space of square integrable functions defined on X, where {a p } ∞ p=1 is a sequence of real numbers.We assume that for almost all x ∈ X ∞ p=0 With these conditions, the symmetric operator A = (A * ) * admits the defect indices (1, 1) [3], . and moreover, we have We denote by L ψ the sub-space of L 2 (X, µ) generated by the sequence {Ψ p (x)} ∞ p=0 .As L ψ is reduced by A, we consider A on L ψ .Then, we have for all f ∈ L ψ [4] with is an analytical function on Π + with |ω (λ)| ≤ 1, ( λ > 0) .We call the function ρ (σ) scalar spectral function of the operator A.
Theorem 3.1.The set P is closed for pointwise convergence.
Proof: Let's ρ n (t) ∈ P and ρ n (σ) −→ ρ (σ) , n → ∞.It is suffice to establish the possibility of the passage to the limit under the integral sign : as n tends to ∞.
However, for all f ∈ D (A) and for all b > 0 In the same way, we show that : By applying the modified theorem of Helly [10], we have for all f ∈ D (A) as n → ∞,or As D (A) = L ψ , this equality is true for all f ∈ L ψ . 2 We will call ρ κ (σ) = ρ (σ) ∈ P orthogonal scalar spectral function if it corresponds to a constant κ, |κ| = 1.
We denote by G 0 the convex hull of these functions : and G = G 0 for the cenvergence in each point of continuity.