We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare the number of Jordan chains of length at least n corresponding to some eigenvalue to each other. In the operator case, it was recently proved that the difference...
We study matrix pencils sE-A using the associated linear subspace ker[A,-E]. The distance between subspaces is measured in terms of the gap metric. In particular, we investigate the gap distance of a regular matrix pencil to the set of singular pencils and provide upper and lower bounds for it. A relation...
We show that the Kronecker canonical form (which is a canonical decomposition for pairs of matrices) is the representation of a linear relation in a finite dimensional space. This provides a new geometric view upon the Kronecker canonical form. Each of the four entries of the Kronecker canonical form...
This survey article contains various aspects of the direct and inverse spectral problem for twodimensional Hamiltonian systems, that is, two dimensional canonical systems of homogeneous differential equations of the form
Jy'(x) = -zH(x)y(x); x ∈ [0;L); 0 < L ≤ ∞; z ∈ C;
with a real non-negative definite...
We study two-dimensional Hamiltonian systems of the form (•) y'(x) = zJH(x)y(x); x ∈ [s-; s+), where the Hamiltonian H is locally integrable on [s-; s+) and nonnegative, and J := (0 -1 | 1 0). The spectral theory of the equation changes depending on the growth of H towards the endpoint s+; the classical...
The theory of closed sesquilinear forms in the non-semibounded situation exhibits some new features, as opposed to the semibounded situation. In particular, there can be more than one closed form associated with the generalized Friedrichs extension SF of a non-semibounded symmetric operator S (if SF...
In many examples of de Branges spaces symmetry appears naturally. Pres-
ence of symmetry gives rise to a decomposition of the space into two
parts, the ‘even’ and the ‘odd’ part, which themselves can be regarded as
de Branges spaces. The converse question is to decide whether a given
space is the...
We consider a Hamiltonian system of the form y0(x) = JH(x)y(x), with a
locally integrable and nonnegative 2_2-matrix valued Hamiltonian H(x).
In the literature dealing with the operator theory of such equations, it is
often required in addition that the Hamiltonian H is trace{normed, i.e.
satis_es...
In the present paper a subclass of scalar Nevanlinna functions
is studied, which coincides with the class of Weyl functions associated to a
nonnegative symmetric operator of defect one in a Hilbert space. This class
consists of all Nevanlinna functions that are holomorphic on (1; 0) and all
those...