On the reproducing kernel of a Pontryagin space of vector valued polynomials

We give necessary and sufficient conditions under which the reproducing kernel of a Pontryagin space of $d \times 1$ vector polynomials is determined by a generalized Nevanlinna pair of $d \times d$ matrix polynomials.Comment: 33 page

W. Stenger's and M.A. Nudelman's results and resolvent formulas involving compressions

In the first part of this note we give a rather short proof of a generalization of Stenger’s lemma about the compression A to H of a self-adjoint operator A in some Hilbert space H= H⊕ H1. In this situation, S: = A∩ A is a symmetry in H with the canonical self-adjoint extension A and the self-adjoint extension A with exit into H. In the second part we consider relations between the resolvents of A and A like M.G. Krein’s resolvent formula, and corresponding operator models

Compressions of Self-Adjoint Extensions of a Symmetric Operator and MG Krein's Resolvent Formula

Let S be a symmetric operator with finite and equal defect numbers in the Hilbert space H. We study the compressions PHAH of the self-adjoint extensions A of S in some Hilbert space HH. These compressions are symmetric extensions of S in H. We characterize properties of these compressions through the corresponding parameter of A in M.G. Kreins resolvent formula. If dim(HH) is finite, according to Stengers lemma the compression of A is self-adjoint. In this case we express the corresponding parameter for the compression of A in Kreins formula through the parameter of the self-adjoint extension A.(VLID)348178

Operators without eigenvalues in finite-dimensional vector spaces

We introduce the concept of a canonical subspace of Cd[z] and among other results prove the following statements. An operator in a finite-dimensional vector space has no eigenvalues if and only if it is similar to the operator of multiplication by the independent variable on a canonical subspace of Cd[z]. An operator in a finite-dimensional Pontryagin space is symmetric and has no eigenvalues if and only if it is isomorphic to the operator of multiplication by the independent variable in a canonical subspace of Cd[z] with an inner product determined by a full matrix polynomial Nevanlinna kernel