Section 23: Applied operator theory - GAMM 2012

Section 23: Applied operator theory 1
Section 23: Applied operator theory
Organizers: Jussi Behrndt (TU Graz), Carsten Trunk (TU Ilmenau)
S23.1: Spectral Theory in Hilbert and Krein Spaces Tue, 13:30–15:30
Chair: Carsten Trunk S1|03–107
Dirac-Krein operators on star graphs
Vadim Adamyan (Odessa National I.I. Mechnikov University)
The talk focuses on the description of the spectrum of a self-adjoint Dirac-Krein differential
operator
0
H = −
1
−1 d
0 dx +
p(x)
q(x)
q(x)
,
−p(x)
on an n-pointed compact star graph Γ, where p(x), q(x) are continuous real-valued functions on
the edges of Γ. The operator H is considered as a perturbation of the orthogonal sum H(12) of the
self-adjoint Dirac-Krein operators on the disjoint edges of Γ, defined on two-component vector
functions with zero first component at one end point and zero second component at the other
end point of each edge of Γ; the domain of H is assumed to consist of all vector functions the
first components of which coincide at the unique vertex of the star graph where all edges touch,
while the boundary conditions at the pendent ends of all edges are the same as for H(12). As a
main tool we use Krein’s resolvent formula for the resolvent kernels (Green’s functions) of H(12)
and H. We prove that the set of common eigenvalues of H and H(12) coincides with the set of
multiple eigenvalues of H(12), but their multiplicities as eigenvalues of H decreases by one. We
also prove that the sets of simple eigenvalues of H and the set of all eigenvalues of H(12) interlace.
The asymptotic behaviour of the number of eigenvalues of H, multiplicities taken into account,
on spectral intervals (−Λ, 0) and (0, Λ) as Λ → ∞ is derived.
The talk is based on a joint work with Heinz Langer and Christiane Tretter.
Spectral functions of products of selfadjoint operators
Tomas Ya. Azizov, Mikhail Denisov (Voronezh State University), Friedrich Philipp (TU Berlin)
Given two possibly unbounded selfadjoint operators A and G such that the resolvent sets of AG
and GA are non-empty, it is shown that the operator AG has a spectral function on R with
singularities if there exists a polynomial p = 0 such that the symmetric operator Gp(AG) is
non-negative. We apply this result to weighted Sturm-Liouville problems.
Variation of discrete spectra of non-negative operators in Krein spaces
Jussi Behrndt (TU Graz), Leslie Leben (TU Ilmenau), Friedrich Philipp (TU Berlin)
Considered is an additive perturbation of a bounded non-negative operator A in a Krein space
with a likewise bounded non-negative operator C from a Schatten-von Neumann ideal of order
p, such that ker C = ker C 2 and 0 is not a singular critical point of C. We show a qualitative
result on the variation of the discrete spectra of the unperturbed and perturbed operator, that
is, given a finite union ∆ of open intervals with 0 /∈ ∆, there exist enumerations (αn) and (βn) of
the discrete eigenvalues of A and B := A + C in ∆ such that
(βn − αn) ∈ ℓ p .
Sign preserving Perturbations of Eigenvalues
Roland Möws (TU Ilmenau), Jussi Behrndt (TU Graz), Carsten Trunk (TU Ilmenau)

2 Section 23: Applied operator theory
We consider two operators A and B which are self-adjoint in a Krein space (K, [·, ·]) and whose
resolvent difference is one-dimensional, i.e.
dim ran (A − λ) −1 − (B − λ) −1 = 1, λ ∈ ρ(A) ∩ ρ(B).
It is well-known that the algebraic eigenspace corresponding to a real discrete eigenvalue of A
(or B), equipped with [·, ·], is a Krein space. The main result is the following: Assume that there
exists a domain Ω ⊂ C in which A (or, equivalently, B) has similar spectral properties as a
definitizable operator and that A satisfies a certain minimality condition. Moreover, let λ1 and λ2
be two discrete eigenvalues of A in Ω ∩ R such that (λ1, λ2) ⊂ ρ(A) and [·, ·] is positive definite on
both ker(A−λ1) and ker(A−λ2). Then there exists a (discrete) eigenvalue µ of B in (λ1, λ2) such
that [·, ·] is not negative definite on the algebraic eigenspace corresponding to µ. In particular, if
µ is a simple eigenvalue with a corresponding eigenvector f, then [f, f] > 0.
The result is applied to a class of Sturm-Liouville problems with an indefinite weight function.
Zeros of Nevanlinna functions with one negative square
Henrik Winkler (TU Ilmenau)
A generalized Nevanlinna function Q(z) with one negative square has precisely one generalized
zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation
defined by Qτ(z) = (Q(z) − τ)/(1 + τQ(z)), τ ∈ R ∪ {∞}, is a generalized Nevanlinna
function with one negative square. Its generalized zero of nonpositive type α(τ) as a function of τ
defines a path in the closed upper halfplane. Various properties of this path are studied in detail.
A perturbation approach to differential operators with indefinite weights
Jussi Behrndt (TU Graz), Friedrich Philipp (TU Berlin), Carsten Trunk (TU Ilmenau)
In many situations differential operators with indefinite weight functions can be regarded as perturbations
of nonnegative selfadjoint operators in Krein spaces. In this talk we first provide an
abstract result on bounded additive perturbations and apply it afterwards to Sturm-Liouville and
second order elliptic partial differential operators with indefinite weights on unbounded domains.
S23.2: Partial Differential Operators Tue, 16:00–18:00
Chair: Sergey Belyi S1|03–107
Weak Neumann implies Stokes
Horst Heck (TU Darmstadt)
When studying the Navier-Stokes equations, one of the basic models in fluid dynamics, a thorough
understanding of the (linear) Stokes equation is very helpful. In particular, the property
of maximal L p -regularity is a very powerful tool in order to treat the nonlinear equation. In this
presentation we show that the existence of the Helmholtz projection in L q (Ω) is sufficient for the
maximal L p -regularity of the Stokes operator, provided the domain Ω ⊂ R n is smooth enough.
The presented result is a joint work with M. Geissert, M. Hieber, and O. Sawada.
Schrödinger operators with interactions on hypersurfaces
Vladimir Lotoreichik (TU Graz)
In the talk we plan to present a new approach to the definition of self-adjoint Schrödinger operators
with δ and δ ′ interactions on hypersurfaces. This approach uses operator extension theory via quasi

Section 23: Applied operator theory 3
boundary triples. Within our approach we prove results concerning spectral and scattering theory
of Schrödinger operators with interactions on hypersurfaces.
A comparison with the approach based on quadratic forms will be given. The quadratic form
for δ ′ interactions was not constructed so far and the question of its construction was posed as
an open problem by Pavel Exner in 2008. In the talk it will be also presented our solution of that
problem.
The talk is based on a joint work with Jussi Behrndt (Graz University of Technology) and
Matthias Langer (University Strathclyde).
Spectra of selfadjoint elliptic differential operators, Robin-to-Dirichlet maps, and an
inverse problem of Calderón type
Jussi Behrndt, Jonathan Rohleder (TU Graz)
In this talk we consider selfadjoint operator realizations of an elliptic differential expression of the
form
n ∂ ∂
Lu = − ajk u + au
∂xj ∂xk
j,k=1
on a bounded or unbounded domain Ω with certain local or nonlocal Robin type boundary conditions.
We will discuss the connections between the behaviour of a corresponding Robin-to-Dirichlet
maps on the boundary of Ω at its discontinuities and the point, absolutely continuous, and singular
continuous spectra of the operator realization. As an application, we present a mild uniqueness
result for the Calderón or Gelfand inverse problem corresponding to L.
Selfadjoint elliptic differential operators on domains with non-compact boundary
Christian Kühn (TU Berlin)
We consider a uniformly elliptic differential expression L of second order on an open set Ω in R n
with a non-compact boundary. We show selfadjointness of a class of realizations of L in L 2 (Ω).
The talk is based on a joint work with J. Behrndt.
Extensible quasi boundary triples and applications
Till Micheler (TU Berlin), Jussi Behrndt (TU Graz)
We study extensions of symmetric operators in Hilbert spaces via a generalization of boundary
triple methods and also discuss applications to elliptic partial differential operators on smooth
and rough domains.
S23.3: Applied Operator Theory and Linear Systems Wed, 13:30–15:30
Chair: Andras Batkai S1|03–107
The Elusive Drude-Born-Fedorov Model for Chiral Electromagnetic Media.
Rainer Picard, Henrik Freymond (TU Dresden)
In a Hilbert space operator setting, covering a comprehensive class of evolutionary equations, as
a particular application various aspects of material laws for Maxwell’s equations are discussed.
In particular, the Drude-Born-Fedorov model for electromagnetic waves in chiral media is investigated
and well-posedness is shown.
Well-posedness and conservativity for linear control systems (Part 1)
Marcus Waurick (TU Dresden)

4 Section 23: Applied operator theory
We discuss a class of linear control problems in a Hilbert space setting. The aim is to show that
these control problems fit in a particular class of evolutionary equations such that the discussion
of well-posedness becomes easily accessible. We exemplify our findings by a system with unbounded
control and observation operator.
Well-posedness and conservativity for linear control systems (Part 2)
Sascha Trostorff (TU Dresden)
Using the results obtained in part 1 of the talk, we study conservativity of a certain class of linear
control problems. For this purpose we require additional regularity properties of our solution
operator in order to allow pointwise evaluations of our solution. We apply the results to the linear
control system with unbounded observation and control operators mentioned in the first part of
the talk.
On energy conditions for electromagnetic diffraction by apertures
Matthias Kunik (Universität Magdeburg), Norbert Gorenflo (TFH Berlin)
The diffraction of light is considered for a plane screen with an open bounded aperture. The
corresponding solution behind the screen is given explicitly in terms of the Fourier transforms
of the tangential components of the electric boundary field on the screen. All components of the
electric as well as the magnetic field vector are considered. We introduce solutions with global
finite energy behind the screen and describe them in terms of two boundary potential functions.
This new approach leads to a decoupling of the vectorial boundary equations on the screen in the
case of global finite energy. For the physically admissible solutions, i.e. the solutions with local
finite energy, we derive a characterisation in terms of the electric boundary fields.
Approximation methods for a class of perturbed paired convolution equations
Michał A. Nowak (AGH University of Science and Technology, Krakow)
We consider approximation methods for some class of perturbed paired convolution equations (or,
in general, singular equations). Effective error estimates, and simultaneously, decaying properties
for solutions are obtained in terms of some smooth spaces. The talk is based on a joint work with
Petru A. Cojuhari.
Hamiltonians and Riccati equations for unbounded control and observation operators
Christian Wyss, Birgit Jacob (Universität Wuppertal), Hans Zwart (University of Twente, The
Netherlands)
We consider the control algebraic Riccati equation
A ∗ X + XA − XBB ∗ X + C ∗ C = 0
for the case that A is normal with compact resolvent, B ∈ L(U, H−s) and C ∈ L(Hs, Y ), 0 ≤ s ≤ 1.
Here Hs ⊂ H ⊂ H−s are the usual fractional domain spaces corresponding to A. Under certain
additional assumptions on A, B and C we show the existence of infinitely many solutions X of
the Riccati equation using invariant subspaces of the Hamiltonian operator matrix
∗
A −BB
T =
.
−C ∗ C −A ∗
The first problem is here to make sense of T as an operator on H × H, because BB ∗ and C ∗ C
map from Hs to H−s. Our main tools are then Riesz bases of eigenvectors of T and indefinite
inner products. In general the solutions X will be unbounded, but we also obtain conditions for
bounded solutions.

Section 23: Applied operator theory 5
S23.4: Applied Operator Theory Wed, 16:00–18:00
Chair: Jussi Behrndt S1|03–107
Shape Preservation of Evolution Equations
András Bátkai (Loránd Eötvös University Budapest)
Motivated by positivity-, monotonicity-, and convexity preserving differential equations, we introduce
a definition of shape preserving operator semigroups and analyze their fundamental properties.
In particular, we prove that the class of shape preserving semigroups is preserved by
perturbations and taking limits. These results are applied, among others, to partial delay differential
equations.
Sectorial realizations of Stieltjes functions
Sergey Belyi (Troy University)
A class of Stieltjes functions with a special condition is considered. We show that a function
belonging to this class can be realized as the impedance function of a singular L-system with
a sectorial state-space operator. We provide an additional condition on a given function from
this class so that the state-space operator of the realizing L-system is α-sectorial with the exact
angle of sectoriality α. Then these results are applied to L-systems based upon a non-self-adjoint
Schrödinger operator.
The talk is based on a joint work with Yu. Arlinskiĭ and E. Tsekanovskiĭ.
On trace norm estimates
Johannes Brasche (TU Clausthal)
Let E and P be nonnegative quadratic forms in a Hilbert space H and suppose that E and the
sum E + bP is densely defined and closed for every b > 0. Let Hb be the selfadjoint operator
associated to E + bP . We present estimates for the trace norm of
(Hb + 1) −1 − lim
b ′ (Hb ′ + 1)−1
−→∞
In particular, we present a criterion in order that these trace norms tend to zero with maximal
rate, i.e. as fast as O(1/b). We illustrate our results with the aid of point interaction Hamiltonians.
On determining the domain of the adjoint operator
Michal Wojtylak (Jagiellonian University, Cracow)
A theorem that is of aid in computing the domain of the adjoint operator will be presented. It may
serve e.g. as a criterion for selfadjointness of a symmetric operator, for normality of a formally
normal operator or for H–selfadjointness of an H–symmetric operator.
Parabolic Variational and Quasi-Variational Inequalities with Gradient Constraints
Carlos N. Rautenberg (Universität Graz), Michael Hintermüller (HU Berlin)
A class of nonlinear parabolic quasi-variational inequality (QVI) problems with gradient type
constraints in function space is considered. Problems of this type arise, for instance, in the mathematical
modelization of superconductors and elasto-plasticity. The paper addresses existence,
regularity and approximation results based on monotone operator theory, Mosco convergence and
C0 semigroup methods. Numerical tests involving the p-Laplacian operator with several nonlinear
constraints are provided.

6 Section 23: Applied operator theory
On a class of quadratic operator pencils with normal coefficients
Friedrich Philipp (TU Ilmenau), Vladimir Strauss (Universidad Simón Bolívar, Caracas),
Carsten Trunk (TU Ilmenau)
A standard description of damped small oscillations of a continuum or of small oscillations of a
pipe carrying steady-state fluid is done via
T ¨z + R ˙z + V z = 0, (1)
where z is a function with values in a Hilbert space and V and R are unbounded operators. A
classical approach is to investigate solutions of the form u(t) = e tλ φ0, and to transform, under
some additional assumptions, the equation in (1) into
L(λ)φ0 := (λ 2 I + λE + F )φ0 = 0
with bounded operators E and F . We will investigate the operator polynomial L with the coefficients
E = AC and F = C 2 , where C is a bounded normal operator in a Hilbert space H and A
is a bounded selfadjoint operator which commutes with C. If there exists a bounded operator Z1
which is an operator root, i.e., a solution of
Z 2 + ACZ + C 2 = 0,
then L(λ) = λ 2 + λAC + C 2 decomposes into linear factors
L(λ) = (λI − Z1)(λI − Z1),
where Z1 = −AC − Z1.
In our talk we will give sufficient conditions for the existence of an operator root. For this we
will investigate the companion operator (or linearizer) of the operator polynomial L which turns
out to be a normal operator in some Krein space. We will then apply recent results from the
spectral theory of normal operators in Krein spaces.