2021

- Group testing, Steiner Systems, and Reed-Solomon Codes. Mittagsseminar Kombinatorik, Hagen, 2021–02–18. [URL]
- On a minimax principle in spectral gaps. Seminar on Differential Equations and Numerical Analysis, University of Zagreb, Croatia, April 26, digital.

This talk deals with a minimax principle for eigenvalues in gaps of the essential spectrum of perturbed self-adjoint operators. This builds upon an abstract minimax principle devised by Griesemer, Lewis, and Siedentop and recents developments on block diagonalization of operators and forms in the off-diagonal perturbation setting. The Stokes operator is revisited as an example.

- On a minimax principle in spectral gaps. Croatian German meeting on analysis and mathematical physics, Dortmund, March 22–25, slides only. [URL]

This talk deals with a minimax principle for eigenvalues in gaps of the essential spectrum of perturbed self-adjoint operators. This builds upon an abstract minimax principle devised by Griesemer, Lewis, and Siedentop and recents developments on block diagonalization of operators and forms in the off-diagonal perturbation setting. The Stokes operator is revisited as an example.

2020

- On a minimax principle in spectral gaps. DMV Jahrestagung 2020, Minisymposium Spectral theory of operators and matrices and partial differential equations, Chemnitz, September 14–17, digital.

This talk deals with a minimax principle for eigenvalues in gaps of the essential spectrum of perturbed self-adjoint operators. This builds upon an abstract minimax principle devised by Griesemer, Lewis, and Siedentop and recents developments on block diagonalization of operators and forms in the off-diagonal perturbation setting. The Stokes operator is revisited as an example.

- Eigenvalue Lifting for Divergence-Type Operators. DMV Jahrestagung 2020, Minisymposium Spectral theory of operators and matrices and partial differential equations, Chemnitz, September 14–17, digital.

This talk deals with eigenvalue lifting for divergence-type operators which describes the phenomenon that certain eigenvalues are strictly increasing when the second order term is perturbed by some non-negative function with small support. Applications include, e.g., the theory of random divergence-type operators. Since here, the random perturbation affects the coefficients of the second order term, one needs exact knowledge of the dependence on some parameters which are less relevant when working with additive random potentials. The results discussed build upon recent joint work with Ivan Veselić.

- Anderson localization beyond regular Floquet eigenvalues. Сonference on Spectral Theory and Mathematical Physics, Sochi, Russia, February 3–7.

We prove that Anderson localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in $L^2(\mathbb{R}^d)$ is universal. By this we mean that Anderson localization holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. This talk is based on joint work with M. Täufer.

2019

- Unique continuation for the gradient and applications. Oberseminar Analysis, TU Dresden, December 5.
- Wegner estimates for random divergence-type operators. Research Seminar Analysis, Stochastics and Mathematical Physics, TU Chemnitz, December 4.
- The reflection principle in the control problem of the heat equation. 6th Najman Conference on Spectral Theory and Differential Equations, Sveti Martin na Muri, Croatia, September 8–13.

We consider the control problem for the generalized heat equation for a Schrödinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts. Moreover, we show that the corresponding control cost does not exceed the one on the whole domain. This talk is based on joint work with M. Egidi.

- Unique continuation for the gradient and applications. 6th Najman Conference on Spectral Theory and Differential Equations, Sveti Martin na Muri, Croatia, September 8–13, poster.

We present a unique continuation estimate for the gradient of eigenfunctions of $H = −\mathrm{div}A\nabla$, where $A(x)$ is a symmetric, uniformly elliptic matrix. This allows us to derive a Wegner estimate for random divergence type operators of the form $H_\omega = −\mathrm{div}(1 + V_\omega )\mathrm{Id}\nabla$. Here $V_\omega$ is some appropriately chosen, non-negative random field with small support.

- Zufällige Divergenz-Typ Operatoren. Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, TU Dortmund, May 21.

Wir betrachten zufällige Operatoren der Form $H_\omega = −\mathrm{div}(1 + V_\omega )\mathrm{Id}\nabla$. Dabei ist $V_\omega$ ein geeignet gewähltes, nicht-negatives, zufälliges Potential mit kleinem Träger.

- The reflection principle in the control problem of the heat equation. International Conference on Elliptic and Parabolic Problems, Minisymposium Control of Partial Differential Equations, Gaeta, Italy, May 20–24.

We consider the control problem for the generalized heat equation for a Schrodinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts. Moreover, we show that the corresponding control cost does not exceed the one on the whole domain. This talk is based on joint work with M. Egidi.

- Lokalisierung an Bandkanten für nicht-ergodische zufällige Schrödingeroperatoren. Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, TU Dortmund, May 14.

2018

- Approximation durch Ausschöpfungen für das Kontrollproblem der Wärmeleitungsgleichung auf unbeschränkten Gebieten. Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, TU Dortmund, November 20.
- A minimax principle in spectral gaps. Kolloquium für Angewandte Mathematik, TU Hamburg-Harburg, June 28.

In [Doc. Math. 4 (1999),275--283], Griesemer, Lewis, and Siedentop devised an abstract minimax principle for eigenvalues in spectral gaps of perturbed self-adjoint operators. We show that this minimax principle can be adapted to the particular situation of bounded additive perturbations with hypotheses that are easier to check in this case. The latter is demonstrated in the framework of the Davis-Kahan $\sin2\Theta$ theorem. This talked is based on joint work with I. Nakić, M. Täufer, M. Tautenhahn, and I. Veselić.

- A minimax principle in spectral gaps. 40. Nordwestdeutsches Funktionalanalysis Kolloquium, Wuppertal, June 23.

In [Doc. Math. 4 (1999),275--283], Griesemer, Lewis, and Siedentop devised an abstract minimax principle for eigenvalues in spectral gaps of perturbed self-adjoint operators. We show that this minimax principle can be adapted to the particular situation of bounded additive perturbations with hypotheses that are easier to check in this case. The latter is demonstrated in the framework of the Davis-Kahan $\sin2\Theta$ theorem. This talked is based on joint work with I. Nakić, M. Täufer, M. Tautenhahn, and I. Veselić.

- A minimax principle in spectral gaps. Oberseminar Stochastik/Mathematische Physik, University of Hagen, April 11.

In [Doc. Math. 4 (1999),275--283], Griesemer, Lewis, and Siedentop devised an abstract minimax principle for eigenvalues in spectral gaps of perturbed self-adjoint operators. We show that this minimax principle can be adapted to the particular situation of bounded additive perturbations with hypotheses that are easier to check in this case. The latter is demonstrated in the framework of the Davis-Kahan $\sin2\Theta$ theorem. This talked is based on joint work with I. Nakić, M. Täufer, M. Tautenhahn, and I. Veselić.

2017

- A critical example in the subspace perturbation problem. Aspect 17: Asymptotic Analysis and Spectral Theory, Trier, September 25–29.

The variation of closed subspaces associated with isolated components of the spectrum of linear self-adjoint operators under a bounded off-diagonal perturbation is considered. This is studied in terms of the difference of the corresponding orthogonal projections. Although the situation is quite well understood under certain additional assumptions on the spectrum of the unperturbed operator, the general case still poses a lot of unsolved questions. We discuss a finite dimensional example indicating that the general case indeed has a different nature than the situation with the additional spectral assumptions.

- On the subspace perturbation problem. 5th Najman Conference on Spectral Theory and Differential Equations, Opatija, Croatia, September 10–15.

The variation of closed subspaces associated with isolated components of the spectra of linear self-adjoint operators under a bounded additive perturbation is considered. Of particular interest is the least restrictive condition on the norm of the perturbation that guarantees that the maximal angle between the corresponding subspaces is less than $\pi/2$. This problem has been discussed by different authors and is still unsolved in full generality. We give a survey on the most common approaches and recent developments.

- Invariant graph subspaces and block diagonalization. Oberseminar Angewandte Analysis und Mathematische Physik, LMU München, July 26.

The problem of decomposition for unbounded self-adjoint $2\times 2$ block operator matrices by a pair of orthogonal graph subspaces is discussed. As a byproduct of our consideration, a new block diagonalization procedure is suggested that resolves related domain issues. The results are discussed in the context of a two-dimensional Dirac-like Hamiltonian. The talk is based on joint work with Konstantin A. Makarov and Stephan Schmitz.

- Invariant graph subspaces and block diagonalization. Oberseminar Analysis, Mathematische Physik & Dynamische Systeme, TU Dortmund, January 17.

The problem of decomposition for unbounded self-adjoint $2\times 2$ block operator matrices by a pair of orthogonal graph subspaces is discussed. As a byproduct of our consideration, a new block diagonalization procedure is suggested that resolves related domain issues. The results are discussed in the context of a two-dimensional Dirac-like Hamiltonian. The talk is based on joint work with Konstantin A. Makarov and Stephan Schmitz.

2016

- On the subspace perturbation problem. Spectral Theory, Differential Equations and Probability, Trilateral German-Russian-Ukrainian summer school, Mainz, September 4–15.

The variation of closed subspaces associated with isolated components of the spectra of linear self-adjoint operators under a bounded additive perturbation is considered. Of particular interest is the least restrictive condition on the norm of the perturbation that guarantees that the maximal angle between the corresponding subspaces is less than $\pi/2$. This problem has been discussed by different authors and is still unsolved in full generality. We give a survey on the most common approaches and recent developments.

2015

- On an estimate in the subspace perturbation problem. Young Researchers Workshop on Spectral Theory, Bern, Switzerland, October 28–30.

We study the problem of variation of spectral subspaces for linear self-adjoint operators under an additive perturbation. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators in terms of the strength of the perturbation. In our approach, we formulate a constrained optimization problem on a finite set of parameters, whose solution gives an estimate on the norm of the difference of the corresponding spectral projections. In particular, this estimate is stronger than the one recently obtained by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 1389–1416].

- On the subspace perturbation problem. Aspect 15: Asymptotic Analysis and Spectral Theory, Orsay, France, October 5–7.

The variation of closed subspaces associated with isolated components of the spectra of linear self-adjoint operators under a bounded additive perturbation is considered. Of particular interest is the least restrictive condition on the norm of the perturbation that guarantees that the maximal angle between the corresponding subspaces is less than $\pi/2$. This problem has been discussed by different authors and is still unsolved in full generality. We give a survey on the most common approaches and recent developments.

2014

- On the subspace perturbation problem. Functional Analysis, Operator Theory and Applications, Workshop on the Occasion of the 90th Birthday of Professor Heinz Günther Tillmann, Mainz, October 23–25.

2012

- On an estimate in the subspace perturbation problem. Spectral Theory and Differential Operators, Graz, Austria, August 27–31.

We study the problem of variation of spectral subspaces for linear self-adjoint operators under an additive perturbation. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators in terms of the strength of the perturbation. In our approach, we formulate a constrained optimization problem on a finite set of parameters, whose solution gives an estimate on the norm of the difference of the corresponding spectral projections. In particular, this estimate is stronger than the one recently obtained by Albeverio and Motovilov in [arXiv:1112.0149v2 (2011)].

- On an estimate in the subspace perturbation problem. Aspect 12: Asymptotic Analysis and Spectral Theory on Non-Compact Structures, Mainz, September 10–12.

We study the problem of variation of spectral subspaces for linear self-adjoint operators under an additive perturbation. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators in terms of the strength of the per-turbation. In our approach, we formulate a constrained optimization problem on a finite set of parameters, whose solution gives an estimate on the norm of the difference of the corresponding spectral projections. In particular, this estimate is stronger than the one recently obtained by Albeverio andMotovilov in [arXiv:1112.0149v2 (2011)].

2011

- A new estimate in the subspace perturbation problem. 3rd St.Petersburg Conference in Spectral Theory, dedicated to the memory of M.Sh.Birman, Saint-Petersburg, Russia, July 1–6.

We study the problem of variation of spectral subspaces for bounded linear self-adjoint operators. We obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. This improves a result obtained earlier by Kostrykin, Makarov and Motovilov in [Trans. Amer. Math. Soc. (2007)]. This talk is based on a joint work with K. A. Makarov.

- A new estimate in the subspace perturbation problem. Operator theory & boundary value problems, Orsay, France, May 25–27.

We study the problem of variation of spectral subspaces for bounded linear self-adjoint operators. We obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. This improves a result obtained earlier by Kostrykin, Makarov and Motovilov in [Trans. Amer. Math. Soc. (2007)]. This talk is based on a joint work with K. A. Makarov.

2010

- A new estimate in the subspace perturbation problem. Analysis and Probability Seminar, Chalmers, Gothenburg, Sweden, November 9.

We study the problem of variation of spectral subspaces for bounded linear self-adjoint operators in a (separable) Hilbert space. We obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators, respectively. In particular, recent results by Kostrykin, Makarov and Motovilov from [Trans. Amer. Math. Soc., V. 359, No. 1, 77--89] and [Proc. Amer. Math. Soc., 131, 3469--3476] are strengthened. This talk is based on a joint work with K. A. Makarov.

- A new estimate in the subspace perturbation problem. IWOTA 2010: 21st International Workshop on Operator Theory and its Applications, Special Session Quantitative Spectral Theory of Block Matrix Operators, Berlin, July 12–16.

We study the problem of variation of spectral subspaces for bounded linear self-adjoint operators. We obtain a new estimate on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators. The result is discussed for off-diagonal perturbations. In this case this improves a result obtained earlier by Kostrykin, Makarov and Motovilov in [Trans.Amer. Math. Soc. (2007)]. This talk is based on a joint work with K. A. Makarov.

TU Dortmund

Fakultät für Mathematik

Lehrstuhl IX

Vogelpothsweg 87

44227 Dortmund

Sie finden uns auf dem sechsten Stock des Mathetowers.

Janine Textor (Raum M 620)

Tel.: (0231) 755-3063

Fax: (0231) 755-5219

Mail: janine.textor@tu-dortmund.de