13 documents found

Article / Chapter
All rights reserved
2018-06-22

Finite rank perturbations of linear relations and singular matrix pencils

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...
Article / Chapter
All rights reserved
2018-03-22

The gap distance to the set of singular matrix pencils

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...
Article / Chapter
All rights reserved
2015-08-03

Linear relations and the Kronecker canonical form

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...
Article / Chapter
All rights reserved
2013-11-22

Two-dimensional Hamiltonian systems

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...
Article / Chapter
All rights reserved
2013-11-22

A growth condition for Hamiltonian systems related with Krein strings

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...
Article / Chapter
All rights reserved
2013-11-22

Non-semibounded closed symmetric forms associated with a generalized Friedrichs...

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...
Article / Chapter
All rights reserved
2013-04-11

Symmetry in de Branges almost Pontryagin spaces

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...
Article / Chapter
All rights reserved
2011-12-13

Reparametrizations of non trace-normed Hamiltonians

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...
Article / Chapter
All rights reserved
2011-12-13

Linear fractional transformations of Nevanlinna functions associated with...

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...