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Abstracts of WK As Weihnachtskolloquium 2016 Vienna University of Technology, December 20, 2016 Participants & Abstracts List of Talks Eckhardt Jonathan On the inverse spectral method for solving the Camassa{Holm equation . 1 Tuesday 15'00 (HS 7) Gantner Jonathan Fractional powers of quaternionic linear operators . 1 Tuesday 17'00 (HS 7) Gerhat Borbala The Krein-Rutman Theorem . 1 Mercedes Tuesday 12'00 (HS 7) Neuner Christoph Super Singular Perturbations for Non-Semibounded Self-Adjoint Operators . 2 Tuesday 17'30 (HS 7) Pruckner Raphael Estimates for order of Nevanlinna matrices . 2 Tuesday 15'30 (HS 7) Rojik Claudio Superconvergence in finite element methods for Maxwell's equations: A local a priori error estimate . 3 Tuesday 12'30 (HS 7) Schwenninger Felix L1-admissibility and Orlicz spaces . 4 Tuesday 16'30 (HS 7) List of Participants Eckhardt Jonathan (University of Vienna), [email protected] Faustmann Markus (TU Vienna), [email protected] Gantner Jonathan (Politecnico di Milano), [email protected] Gerhat Borbala Mercedes (TU Vienna), [email protected] Neuner Christoph (Stockholms Universitet), [email protected] Pruckner Raphael (TU Vienna), [email protected] Rojik Claudio (TU Vienna), [email protected] Schwenninger Felix (University of Hamburg), [email protected] Abstracts On the inverse spectral method for solving the Camassa{Holm equation Eckhardt Jonathan Tuesday 15'00 (HS 7) The Camassa{Holm equation is a nonlinear partial differential equation that models unidirec- tional wave propagation on shallow water. I will show how to integrate this equation by means of solving an inverse spectral problem for a Sturm{Liouville problem with an indefinite weight. ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ Fractional powers of quaternionic linear operators Gantner Jonathan Tuesday 17'00 (HS 7) In recent years, the fundamental concepts of operator theory have been extended to linear operators on Banach and Hilbert spaces over the skew field of quaternions. The natural gener- alization of the holomorphic functional calculus has been developed and it was even possible to prove the spectral theorem for normal quaternionic linear operators. Two crucial steps in the de- velopment of the theory were the introduction of the S-spectrum, the correct notion of spectrum in this setting, and the identification of slice-hyperholomorphicity as the notion of generalized holomorphicity that underlies quaternionic operator theory. Using the theory of slice hyperholomorphic functions, we were able to generalize further classic results. This talk shows how to construct fractional powers of quaternionic linear operators using the above concepts and discusses several of their properties. In particular we show that an analogue of Kato's famous formula for the resolvent of the fractional power of an operator exists in the quaternionic setting. The talk concludes with an outlook on future work concerning possible applications in frac- tional evolution. ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ The Krein-Rutman Theorem Gerhat Borbala Mercedes Tuesday 12'00 (HS 7) In this talk I present an outline of my master thesis, which discusses the Krein-Rutman Theorem on the spectrum of compact, strongly positive linear operators on real, ordered Banach spaces. It represents an infinite dimensional generalisation of the better known Perron-Frobenius Theorem, which characterizes the spectrum of real matrices with positive entries. Preparing the setting of the theorem, I introduce real, ordered Banach spaces via order cones and strongly positive, linear and bounded operators acting on them. Under certain assumptions, 1 the topological dual space of an ordered Banach space can be equipped with a corresponding order relation. Finally, I present the Krein-Rutman Theorem, which amongst other strong results states that the spectral radius of a strongly positive, compact linear operator on a real, ordered Banach space is a positive and algebraically simple eigenvalue with a corresponding positive eigenvector. ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ Super Singular Perturbations for Non-Semibounded Self-Adjoint Operators Neuner Christoph Tuesday 17'30 (HS 7) Let A be a self-adjoint operator in a Hilbert space H, where the spectrum potentially occupies the whole real line. We are interested to describe the perturbations A+αh'; ·i' for α 2 R[f1g. If ' is from the Hilbert space then this is easily done. If ' is singular, by which we mean it is from the rigged spaces H−1(A) or H−2(A), the description becomes a bit more technical but is still well understood. We are therefore interested in super singular elements ', i.e., such that ' 2 H−n(A) for n ≥ 3. Using an existing Hilbert space model (due to Kurasov) for semibounded self-adjoint operators, we investigate how this model behaves in the non-semibounded case and which, if any, of its key features can be salvaged. This is joint work with Pavel Kurasov and Annemarie Luger. ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ Estimates for order of Nevanlinna matrices Pruckner Raphael Tuesday 15'30 (HS 7) All solutions of an indeterminate Hamburger moment problem can be described with a Nevan- linna matrix. The four entries of this matrix are entire functions with the same exponential order. We are interessted to determine this value. We write the moment problem as a canonical system with Hamiltonian H. Here, H : [0;L) ! R2×2 is a locally integrable function whose values are a.e. positive semi-definite. The correspond- ing canonical system is given by the equation y0(x) = zJH(x)y(x); x 2 (0;L); 0 −1 where z 2 C and J = 1 0 . The so-called Hamburger Hamiltonians which appear here have a much easier structure. We obtain estimates for the order by transforming a given Hamburger Hamiltonian into (the Hamiltonian associated with) a Krein-string, and apply a theorem of I.S.Kac to evaluate the order of that string. Our result an be viewed as a generalisation of a theorem by Berezanskii in the 50s. On the way, we leave the positive definite scheme and encounter Hamiltonians which take also negative definite matrices as values. ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ 2 Superconvergence in finite element methods for Maxwell's equations: A local a priori error estimate Rojik Claudio Tuesday 12'30 (HS 7) Given a domain Ω ⊆ Rn and the bilinear form Z a(u; v) = ru · rv dx Ω corresponding to Poisson's equation −∆u = f for u; v 2 H1(Ω), Nitsche and Schatz [1] showed the following result: Denote by B0 a ball of radius r and by Bd the concentric ball of radius r + d, where d is large 1 compared to the element diameter h of the FEM triangulation. If uh 2 Sh(Ω) is a H -conforming 1 FEM approximation to u 2 H0 (Ω) satisfying a(u − uh; χ) = 0 for all χ 2 Sh(Bd) with compact support in Bd, then −1 1 1 2 ku − uhkH (B0) ≤ C min ku − χkH (Bd) + d ku − χkL (Bd) χ2Sh(Bd) −1 2 + Cd ku − uhkL (Bd): Inspired by this result, we try to establish similar error estimates for Maxwell's equations in the form curl curl E + κE = j; where E denotes the vector potential, j is the given current density and κ is a coefficient de- pending on the setting, with the corresponding variational formulation b(u; v) = (j; v)L2(Ω) for Ω ⊆ R3 and b(u; v) = (curl u; curl v)L2(Ω) + κ(u; v)L2(Ω) as the bilinear form. It seems natural to choose the FEM approximation to be H(curl)- conforming. Within this talk for the 5. Weihnachtskolloquium, we want to present results from our current research to this topic. References [1] J. A. Nitsche and A. H. Schatz. Interior estimates for ritz-galerkin methods. Math. Comp., 28:937-958, 1974. ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ 3 L1-admissibility and Orlicz spaces Schwenninger Felix Tuesday 16'30 (HS 7) In this talk we discuss the boundedness of the linear operator Z 1 Φ: L1(0; 1) 7! X; u 7! u(s)AT (s)x ds; 0 where A is the generator of an analytic semigroup T on the Banach space X and x 2 X. This question can be linked to stability notions in linear systems theory. The talk is based on joint work with B. Jacob, R. Nabiullin and J.R. Partington. ∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼ 4.
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