On the negative squares of a class of self-adjoint extensions in Krein spaces. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 286 (2013), 2/3, S. 118-148
A description of all exit space extensions with finitely many negative squares of a symmetric operator of defect one is given via Krein's formula. As one of the main results an exact characterization of the number of negative squares in terms of a fixed canonical extension and the behaviour of a function $\tau$ (that determines the exit space extension in Krein's formula) at zero and at infinity is obtained. To this end the class of matrix valued $\mathcal D_\kappa^{n\times n}$-functions is introduced and, in particular, the properties of the inverse of a certain $\mathcal D_\kappa^{2\times 2}$-function which is closely connected with the spectral properties of the exit space extensions with finitely many negative squares is investigated in detail. Among the main tools here are the analytic characterization of the degree of non-positivity of generalized poles of matrix valued generalized Nevanlinna functions and some extensions of recent factorization results.
http://dx.doi.org/10.1002/mana.201000154
Local spectral theory for normal operators in Krein spaces. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 286 (2013), 1, S. 42-58
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 normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.
http://dx.doi.org/10.1002/mana.201000141
Some Sobelev spaces as Pontryagin spaces. - In: Vestnik Južno-Ural'skogo Gosudarstvennogo Universiteta. Serija matematika, mechanika, fizika / Južno-Uralьskij gosudarstvennyj universitet. - Čeljabinsk, 2014- , ISSN: 2075-809X , ZDB-ID: 2701282-7, ISSN 2075-809X, Bd. 6.2012, 11 (270), S. 14-23
We show that well known Sobolev spaces can quite naturally be treated as Pontryagin spaces. This point of view gives a possibility to obtain new properties for some traditional objects such as simplest differential operators.
On similarity of indefinite Sturm-Liouville operators. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 12 (2012), 1, S. 771-772
http://dx.doi.org/10.1002/pamm.201210374
On finite rank perturbations of selfadjoint operators in Krein spaces and eigenvalues in spectral gaps. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2012. - Online-Ressource (PDF-Datei: 10 S., 150,6 KB). - (Preprint ; M12,12)
It is shown that the finiteness of eigenvalues in a spectral gap of a definitizable or locally definitizable selfadjoint operator in a Krein space is preserved under finite rank perturbations. This results is applied to a class of singular Sturm-Liouville operators with an indefinite weight function.
http://www.db-thueringen.de/servlets/DocumentServlet?id=20974
The numerical range of non-negative operators in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2012. - Online-Ressource (PDF-Datei: 21 S., 177,7 KB). - (Preprint ; M12,11)
We define and characterize the Krein space numerical range W(A) and the Krein space co-numerical range W_{\rm co}(A) of a non-negative operator A in a Krein space. It is shown that the non-zero spectrum of A is contained in the closure of W(A)\cap W_{\rm co}(A).
http://www.db-thueringen.de/servlets/DocumentServlet?id=20973
Bounds on the non-real spectrum of differential operators with indefinite weights. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2012. - Online-Ressource (PDF-Datei: 27 S., 276,8 KB). - (Preprint ; M12,07)
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 assumption that 0 and infinity are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm-Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.
http://www.db-thueringen.de/servlets/DocumentServlet?id=20509
PT symmetric, Hermitian and P-self-adjoint operators related to potentials in PT quantum mechanics. - In: Journal of mathematical physics, ISSN 1089-7658, Bd. 53 (2012), 1, S. 012109-1-012109-18
https://doi.org/10.1063/1.3677368
Spectral theory, mathematical system theory, evolution equations, differential and difference equations : 21st International Workshop on Operator Theory and Applications, Berlin, July 2010. - Basel : Birkhäuser, 2012. - VIII, 690 S.. - (Operator theory ; 221) ISBN 3-0348-0296-X
Literaturangaben
Local definitizability of T [*]T and TT[*]. - In: Integral equations and operator theory, ISSN 1420-8989, Bd. 71 (2011), 4, S. 491-508
http://dx.doi.org/10.1007/s00020-011-1913-0