Publications (37)
The invertibility of 2X2 operator matrices
Junjie Huang, Junfeng Sun, Alatancang Chen +1
In this paper the properties of right invertible row operators, i.e., of 1X2 surjective operator matrices are studied. This investigation is based on a specific space decomposition…
Local spectral theory for normal operators in Krein spaces
Friedrich Philipp, Vladimir Strauss, Carsten Trunk
Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for…
Perturbation and spectral theory for singular indefinite Sturm-Liouville operators
Jussi Behrndt, Philipp Schmitz, Gerald Teschl +1
We study singular Sturm-Liouville operators of the form \[ \frac{1}{r_j}\left(-\frac{\mathrm d}{\mathrm dx}p_j\frac{\mathrm d}{\mathrm dx}+q_j\right),\qquad j=0,1, \] in $L^2((a,b)…
The spectrum and the Weyr characteristics of operator pencils and linear relations
Hannes Gernandt, Carsten Trunk
The relation between the spectra of operator pencils with unbounded coefficients and of associated linear relations is investigated. It turns out that various types of spectrum coi…
The numerical range of non-negative operators in Krein spaces
Friedrich Philipp, Carsten Trunk
We define and characterize the Krein space numerical range and the Krein space co-numerical range of a non-negative operator in a Krein space. It is show…
Locally finite extensions and Gesztesy-Å eba realizations for the Dirac operator on a metric graph
Hannes Gernandt, Carsten Trunk
We study extensions of direct sums of symmetric operators . In general there is no natural boundary triplet for even if there is one for every…
Eigenvalue estimates for singular left-definite Sturm-Liouville operators
Jussi Behrndt, Roland Moews, Carsten Trunk
The spectral properties of a singular left-definite Sturm-Liouville operator are investigated and described via the properties of the corresponding right-definite selfadjoint…
Indefinite Sturm-Liouville operators in polar form
Branko Äurgus, Volodymyr Derkach, Carsten Trunk
We consider the indefinite Sturm-Liouville differential expression \[\mathfrak{a}(f) := - \frac{1}{w}\left( \frac{1}{r} f' \right)',\] where is defined on a finite o…
Spectral enclosures for a class of block operator matrices
Juan Giribet, Matthias Langer, Francisco MartÃnez PerÃa +2
We prove new spectral enclosures for the non-real spectrum of a class of block operator matrices with self-adjoint operators and on the diagonal and operators $B…
Spectral Points of Type and Type of Closed Operators in Indefinite Inner Product Spaces
Friedrich Philipp, Carsten Trunk
We introduce the notion of spectral points of type and type of closed operators in a Hilbert space which is equipped with an indefinite inner product. It is shown…
On non-negative operators in Krein spaces and their perturbations
Jussi Behrndt, Friedrich M. Philipp, Carsten Trunk
One of the most important contributions of Heinz Langer in the area of operator theory in Krein spaces is the introduction of the notion of definitizable operators and the construc…
On the solvability of boundary value problems for linear differential-algebraic equations with constant coefficients
Anar Assanova, Carsten Trunk, Roza Uteshova
We study a two-point boundary value problem for a linear differen\-tial-algebraic equation with constant coefficients by using the method of parameterization. The parameter is set…
Spectral bounds for singular indefinite Sturm-Liouville operators with --potentials
Jussi Behrndt, Philipp Schmitz, Carsten Trunk
The spectrum of the singular indefinite Sturm-Liouville operator with a real potential covers the…
Feedback rectifiable pairs and stabilization of switched linear systems
Maria C. Honecker, Hannes Gernandt, Kai Wulff +2
We address the feedback design problem for switched linear systems. In particular we aim to design a switched state-feedback such that the resulting closed-loop subsystems share th…
Operator based approach to PT-symmetric problems on a wedge-shaped contour
Florian Leben, Carsten Trunk
We consider a second-order differential equation with an eigenvalue parameter . In quantum mechan…
On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)
Tomas Ya. Azizov, Carsten Trunk
In the recent years a generalization of Hermiticity was investigated using a complex deformation H=p^2 +x^2(ix)^εof the harmonic oscillator Hamiltonian, where εis a real paramete…
Variational principles for self-adjoint operator functions arising from second-order systems
Birgit Jacob, Matthias Langer, Carsten Trunk
Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form \[ \langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z…
Analyticity and Riesz basis property of semigroups associated to damped vibrations
Birgit Jacob, Carsten Trunk, Monika Winklmeier
Second order equations of the form in an abstract Hilbert space are considered. Such equations are often used as a model for transverse motions of thin beams…
On characteristic invariants of matrix pencils and linear relations
Hannes Gernandt, Francisco MartÃnez PerÃa, Friedrich Philipp +1
The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or…
Perturbations of periodic Sturm--Liouville operators
Jussi Behrndt, Philipp Schmitz, Gerald Teschl +1
We study perturbations of the self-adjoint periodic Sturm--Liouville operator \[ A_0 = \frac{1}{r_0}\left(-\frac{\mathrm d}{\mathrm dx} p_0 \frac{\mathrm d}{\mathrm dx} + q_0\right…
Square roots of H-nonnegative matrices
Dawie B. Janse van Rensburg, Madelein van Straaten, Frieda Theron +1
Roots of matrices are well-studied. The conditions for their existence are understood: The block sizes of nilpotent Jordan blocks, arranged in pairs, have to satisfy some simple al…
PT Symmetric, Hermitian and P-Self-Adjoint Operators Related to Potentials in PT Quantum Mechanics
Tomas Azizov, Carsten Trunk
In the recent years a generalization of the harmonic oscillator using a complex deformation was investigated, where ε is a real parameter. Here, we will consid…
Lower bounds for self-adjoint Sturm-Liouville operators
Jussi Behrndt, Fritz Gesztesy, Philipp Schmitz +1
In this note we provide estimates for the lower bound of the self-adjoint operator associated with the three-coefficient Sturm-Liouville differential expression $$ \frac{1}{r} \lef…
On a class of non-Hermitian matrices with positive definite Schur complements
Thomas Berger, Juan Ignacio Giribet, Francisco MartÃnez PerÃa +1
Given a positive definite matrix and a Hermitian matrix , we characterize under which conditions there exists a strictly…
Spectrum of -frame operators
Juan Ignacio Giribet, Matthias Langer, Leslie Leben +3
A -frame is a frame for a Krein space which is compatible with the indefinite inner product in the sense that it induces an…
Hermitian Pencils and their Representation in Krein Spaces
Rabeb Aydi, Omaima Kchaou, Carsten Trunk
Pencils of the form are studied, where and are bounded linear operators on a Hilbert space. Of interest are the spectral properties of $\mathcal{A…
The effect of finite rank perturbations on Jordan chains of linear operators
Jussi Behrndt, Leslie Leben, Francisco MartÃnez PerÃa +1
A general result on the structure and dimension of the root subspaces of a matrix or a linear operator under finite rank perturbations is proved: The increase of dimension from the…
Limit point and limit circle trichotomy for Sturm-Liouville problems with complex potentials
Florian Leben, Edison Leguizamón, Carsten Trunk +1
The limit point and limit circle classification of real Sturm-Liouville problems by H. Weyl more than 100 years ago was extended by A.R. Sims around 60 years ago to the case when t…
A Jordan-like decomposition for linear relations in finite-dimensional spaces
Thomas Berger, Henk de Snoo, Carsten Trunk +1
A square matrix has the usual Jordan canonical form that describes the structure of via eigenvalues and the corresponding Jordan blocks. If is a linear relation in a fi…
Bounds on the non-real spectrum of differential operators with indefinite weights
Jussi Behrndt, Friedrich Philipp, Carsten Trunk
Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumptio…
Eigenvalue placement for regular matrix pencils with rank one perturbations
Hannes Gernandt, Carsten Trunk
A regular matrix pencil sE-A and its rank one perturbations are considered. We determine the sets in the extended complex plane which are the eigenvalues of the perturbed pencil. W…
Linear relations and their singular chains
Thomas Berger, Henk de Snoo, Carsten Trunk +1
Singular chain spaces for linear relations in linear spaces play a fundamental role in the decomposition of linear relations in finite-dimensional spaces. In this paper singular ch…
1-D Schrödinger operator on a star graph with nondefinite weight function
Edison Leguizamón, Carsten Trunk, Mitsuru Wilson +1
On a star graph with edges of unit length, we study the operator on and on $n…
Finite Rank Perturbations of Linear Relations and Matrix Pencils
Leslie Leben, Francisco MartÃnez-PerÃa, Friedrich Philipp +2
We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of…
Numerical Range and Quadratic Numerical Range for Damped Systems
Birgit Jacob, Christiane Tretter, Carsten Trunk +1
We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations $\ddot{z}(t) + D \dot{z} (t) + A_0 z(t) = 0…
Relative oscillation theory and essential spectra of Sturm--Liouville operators
Jussi Behrndt, Philipp Schmitz, Gerald Teschl +1
We develop relative oscillation theory for general Sturm-Liouville differential expressions of the form \[ \frac{1}{r}\left(-\frac{\mathrm d}{\mathrm dx} p \frac{\mathrm d}{\mathrm…
On a class of -self-adjoint operators with empty resolvent set
Sergii Kuzhel, Carsten Trunk
In the present paper we investigate the set of all -self-adjoint extensions of a symmetric operator with deficiency indices which commutes with a non-trivial…