Eigenvalue estimates for singular left-definite Sturm-Liouville operators. - In: Journal of spectral theory, ISSN 1664-0403, Bd. 1 (2011), 3, S. 327-347
The spectral properties of a singular left-definite Sturm-Liouville operator $JA$ are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart $A$ which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the $J$-selfadjoint operator $JA$ is real and it follows that an interval $(a,b)\subset\dR^+$ is a gap in the essential spectrum of $A$ if and only if both intervals $(-b,-a)$ and $(a,b)$ are gaps in the essential spectrum of the $J$-selfadjoint operator $JA$. As one of the main results it is shown that the number of eigenvalues of $JA$ in $(-b,-a) \cup (a,b)$ differs at most by three of the number of eigenvalues of $A$ in the gap $(a,b)$; as a byproduct results on the accumulation of eigenvalues of singular left-definite Sturm-Liouville operators are obtained. Furthermore, left-definite problems with symmetric and periodic coefficients are treated, and several examples are included to illustrate the general results.
https://doi.org/10.4171/JST/14
Selfadjoint operators in S-spaces. - In: Journal of functional analysis, ISSN 1096-0783, Bd. 260 (2011), 4, S. 1045-1059
We study S-spaces and operators therein. An S-space is a Hilbert space with an additional inner product given by $\Skindef := (U\,\cdot\,,-)$, where $U$ is a unitary operator. We investigate spectral properties of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that for each selfadjoint operator $A$ in an S-space we find an inner product which turns $\bez$ into a Krein space and $A$ into a selfadjoint operator therein. As a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in Krein spaces, which provides a different insight into this well know and in general unsolved problem.
https://doi.org/10.1016/j.jfa.2010.10.023
Spectral points of definite type and type π for linear operators and relations in Krein spaces. - In: Journal of the London Mathematical Society, ISSN 1469-7750, Bd. 83 (2011), 3, S. 768-788
Spectral points of positive and negative type, and type $\pi_{+}$ and type $\pi_{-}$ for closed linear operators and relations in Krein spaces are introduced with the help of approximative eigensequences. The main objective of the paper is to study these sign type properties in the non-selfadjoint case under various kinds of perturbations, e.g. compact perturbations and perturbations small in the gap metric. Many of the obtained perturbation results are also new for the special case of bounded and unbounded selfadjoint operators in Krein spaces.
http://dx.doi.org/10.1112/jlms/jdq098
Frequency compensation for a class of DAE's arising in electrical circuits. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 11 (2011), 1, S. 837-838
Structured perturbations of regular pencils of the form $sE-A$, $E,A\in\dR^{n\times n}, ˜s\in\dC,$ are considered which model the addition of a capacitance $c$ in an electrical circuit in order to improve the frequency response.
http://dx.doi.org/10.1002/pamm.201110407
On a class of J-self-adjoint operators with empty resolvent set. - In: Journal of mathematical analysis and applications, ISSN 1096-0813, Bd. 379 (2011), 1, S. 272-289
https://doi.org/10.1016/j.jmaa.2010.12.048
PT symmetric, hermitian and P-self-adjoint operators related to potentials in PT quantum mechanics. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2011. - Online-Ressource (PDF-Datei: 28 S., 189,9 KB). - (Preprint ; M11,15)
In the recent years a generalization H=p^2 + x^2(ix)^\epsilon of the harmonic oscillator using a complex deformation was investigated, where \epsilon is a real parameter. Here, we will consider the most simple case: \epsilon even and x real. We will give a complete characterization of three different classes of operators associated with the differential expression H: The class of all self-adjoint (Hermitian) operators, the class of all PT symmetric operators and the class of all P-self-adjoint operators. Surprisingly, some of the PT symmetric operators associated to this expression have no resolvent set.
http://www.db-thueringen.de/servlets/DocumentServlet?id=18815
Frequency compensation for a class of DAE's arising in electrical circuits. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2011. - Online-Ressource (PDF-Datei: 5 S., 115,2 KB). - (Preprint ; M11,11)
http://www.db-thueringen.de/servlets/DocumentServlet?id=18489
A perturbation approach to differential operators with indefinite weights. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2011. - Online-Ressource (PDF-Datei: 18 S., 210,5 KB). - (Preprint ; M11,08)
http://www.db-thueringen.de/servlets/DocumentServlet?id=18137
Eigenvalue estimates for singular left-definite Sturm-Liouville operators. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2011. - Online-Ressource (PDF-Datei: 15 S., 379,8 KB). - (Preprint ; M11,01)
The spectral properties of a singular left-definite Sturm-Liouville operator JA are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart A which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the J-selfadjoint operator JA is real and it follows that an interval (a; b) \subset {\Bbb R}+ is a gap in the essential spectrum of A if and only if both intervals (-b;-a) and (a; b) are gaps in the essential spectrum of the J-selfadjoint operator JA. As one of the main results it is shown that the number of eigenvalues of JA in (-b;-a) [ (a; b) di ers at most by three of the number of eigenvalues of A in the gap (a; b); as a byproduct results on the accumulation of eigenvalues of singular left-definite Sturm-Liouville operators are obtained. Furthermore, left-definite problems with symmetric and periodic coefficients are treated, and several examples are included to illustrate the general results
http://www.db-thueringen.de/servlets/DocumentServlet?id=17288
Small perturbations of selfadjoint and unitary operators in Krein spaces. - In: Journal of operator theory, ISSN 0379-4024, Bd. 64 (2010), 2, S. 401-416
We investigate the behaviour of the spectrum of selfadjoint operators in Krein spaces under perturbations with uniformly dissipative operators. Moreover we consider the closely related problem of the perturbation of unitary operators with uniformly bi-expansive. The obtained perturbation results give a new characterization of spectral points of positive type and of type $\pi_{+}$ of selfadjoint (resp.\ unitary) operators in Krein spaces.