Symmetric relations on a Hilbert space

Self-adjoint extensions of symmetric subspaces

“…Fundamental is that for a self-ad joint subspace H in ξ> 2 the linear manifold JB(λ) = (H -λ) -1 (λ e C -R) is a bounded linear operator defined on all of φ, with the properties i2(λ)* = R(X) and R(x) -R(μ) = (λ -μ)R(X) R(μ). This fact, due to Coddington [5] and also proved by Bennewitz [3], forms the basis of our paper. As was shown by McKelvey [6] and Schneider [11], these relations are sufficient to guarantee the existence of a spectral family E(t) (t e R) such that (*) jβ(λ) -( ---dE(t), λe C -R .…”

Boundary relations and their Weyl families

“…[5], [10]. A symmetric linear relation S is called simple if there is no nontrivial orthogonal decomposition of the Hilbert space H = H 1 ⊕ H 2 and no corresponding orthogonal decomposition S = S 1 ⊕ S 2 with S 1 a symmetric relation in H 1 and S 2 a selfadjoint relation in H 2 .…”

On the solution of Nevanlinna Pick problem with selfadjoint extensions of symmetric linear relations in Hilbert space

“…We may see that the description of Coddington for all selfadjoint extensions in possibly larger spaces is based on the corresponding results for unitary extensions of isometric operators, see [6]. 3. SOLUTION OF THE NEVANLINNA PICK PROBLEM Our interest will be in selfadjoint extensions of a given symmetric subspace (closed linear relation) and this has been discussed, for instance see [3], [5], and [6], when one of that extensions is minimal and its connection with Resolvent has been discussed in [9], [10, [12] and [13].…”

“…{f, f} 7/2} identity operator on 7/2, (P(g) Q(g)) a Nevanlinna pair, for more detail related to (P(g)Q(g)), we refer to [8], [9], [10] We refer to [2], [3], [4] and [5] [6]. In this case the compressed resolvent R (g) of .4 in is called a generalized resolvent of S, or the generalized resolvent of S associated with .4, see [6] and [7].…”

“…3. SOLUTION OF THE NEVANLINNA PICK PROBLEM Our interest will be in selfadjoint extensions of a given symmetric subspace (closed linear relation) and this has been discussed, for instance see [3], [5], and [6], when one of that extensions is minimal and its connection with Resolvent has been discussed in [9], [10, [12] and [13]. Then taking into consideration [5] prs (P(0Q()).…”