Jacob, Birgit; Tretter, Christiane; Trunk, Carsten; Vogt, Hendrik;
Numerical range and quadratic numerical range for damped systems - Ilmenau : Technische Universität, Institut für Mathematik, 2017 - 1 Online-Ressource (27 Seiten). . - (Preprint. - M17,05)

We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations z'' (t) + D z' (t) + A_0 z(t) = 0 in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive and may have the same strength as A_0. By means of the quadratic numerical range, we establish tight spectral estimates in terms of the unbounded operator coefficients A_0 and D which improve earlier results for sectorial and selfadjoint D; in contrast to numerical range bounds, our enclosures may even provide bounded imaginary part of the spectrum or a spectral free vertical strip. An application to small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid illustrates that our new bounds are explicit.

https://www.db-thueringen.de/receive/dbt_mods_00031984

Derkach, Vladimir; Trunk, Carsten;
Coupling of definitizable operators in Krein spaces - Ilmenau : Technische Universität, Institut für Mathematik, 2017 - 1 Online-Ressource (18 Seiten). . - (Preprint. - M17,03)

Indefinite Sturm-Liouville operators defined on the real line are often considered as a coupling of two semibounded symmetric operators defined on the positive and the negative half axis. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension. In this paper we present an abstract approach to the coupling of two (definitizable) self-adjoint operators. We obtain a characterization for the definitizability and the regularity of the critical points. Finally we study a typical class of indefinite Sturm-Liouville problems on the real line.

https://www.db-thueringen.de/receive/dbt_mods_00031469

Giribet, Juan; Langer, Matthias; Leben, Leslie; Maestripieri, Alejandra; Martínez Pería, Francisco; Trunk, Carsten;
Spectrum of J-frame operators - Ilmenau : Technische Universität, Institut für Mathematik, 2017 - 1 Online-Ressource (20 Seiten). . - (Preprint. - M17,01)

A J-frame is a frame F for a Krein space which is compatible with the indefinite inner product in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in H . With every J-frame the so-called J-frame operator is associated, which is a self-adjoint operator in the Krein space H. The J-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of J-frame operators in a Krein space by a 2X2 block operator representation. The J-frame bounds of F are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the 2X2 block representation. Moreover, this 2X2 block representation is utilized to obtain enclosures for the spectrum of J-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all J-frames associated with a given J-frame operator.

https://www.db-thueringen.de/receive/dbt_mods_00031058

Braun, Philipp; Faulwasser, Timm; Grüne, Lars; Kellett, Christopher M.; Weller, Steven R.; Worthmann, Karl;
Maximal islanding time for microgrids via distributed predictive control. - In: 22nd International Symposium on Mathematical Theory of Networks and Systems : July 12-15, 2016, University of Minnesota, Twin Cities Campus. - Minneapolis, MN : University Digital Conservancy, ISBN 978-1-5323-1358-5, (2016), S. 652-659

http://hdl.handle.net/11299/181518

Shkalikov, A. A.; Trunk, Carsten;
On stability of closedness and self-adjointness for 2 x 2 operator matrices. - In: Mathematical notes - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1573-8876, Bd. 100 (2016), 5, S. 870-875

http://dx.doi.org/10.1134/S0001434616110274