Abstract:

We use different types of H-distributions to obtain possible strong convergence of a weakly convergent sequences in different spaces of functions (Lebesgue, Sobolev, Bessel...). This problem is of special interest when we investigate convergence of a sequence of approximate solutions to different classes of PDEs.

Abstract:

Many cases are known where a semigroup has a continuous kernel. We will present a new and very efficient characterization for such kernel to exist. This will be used to study positive semigroups on spaces of continuous functions. Examples are the Robin Laplacian but also the Dirichlet-to-Neumann operator.
A particular challenge is irreducibility.
Ultracontractivity plays a particular role, not only for the kernel but also for the asymptotic behavior of the eigenvalues. Weyl's formula holds for the Dirichlet-to-Neumann operator associated to the Laplacian and also the Laplacian perturbed by a potential. The talk is based on joint work with Tom ter Elst.

References:

W. Arendt, T. ter Elst: The Dirichlet-to-Neumann operator on spaces of continuous functions.
arXiv:1707.05556

W. Arendt, T. ter Elst: Ultracontractivity and Eigenvalues. Weyl's Law for the Dirichlet-to-Neumann Operator. Integral Equations and Operator Theory 88 (2017) 65-89

Abstract:

Consider a system of forward-backward evolution linear equations
\[
\begin{pmatrix} x'(t) \\ y'(t) \end{pmatrix}=
\begin{pmatrix} A & -B \\ -C & -A^* \end{pmatrix}
\begin{pmatrix} x(t) \\ y(t) \end{pmatrix},\quad
\begin{aligned} x(0) &=x_0 \\ y(T) &=G x(T) \end{aligned}\quad
t\in[0,T].
\]
Here $A$ is an accretive (unbounded) operator, $B$, $C$, $G$ are
self-adjoint (unbounded) non-negative operators. This kind of system arises, for example,
in some optimal control problems

It is proved the solvability of the system and the solvability of the related
differential operator Riccati equation in the collection of Banach spaces.

N.V. Artamonov, On the solvability of a system of forward-backward linear equations with unbounded operator coefficients.

Abstract:

We study unimodular transformations of conservative $L$-systems. Classes $\mathfrak M^Q$,
$\mathfrak M^Q_\kappa$, $\mathfrak M^{-1,Q}_\kappa$ that are impedance functions of the corresponding
$L$-systems are introduced. A unique unimodular transformation of a given $L$-system with impedance
function from the mentioned above classes is found such that the impedance function of a new $L$-system
belongs to $\mathfrak M^{(-Q)}$, $\mathfrak M^{(-Q)}_\kappa$, $\mathfrak M^{-1,(-Q)}_\kappa$,
respectively. As a result we get that considered classes (that are perturbations of the Donoghue classes
of Herglotz-Nevanlinna functions with an arbitrary real constant $Q$) are invariant under the
corresponding unimodular transformations of $L$-systems. We define a coupling of an $L$-system and a so
called $F$-system and on its basis obtain a multiplication theorem for their transfer functions. In
particular, it is shown that any unimodular transformation of a given $L$-system is equivalent to a
coupling of this system and the corresponding controller, an $F$-system with a constant unimodular
transfer function.

In addition, we derive an explicit form of a controller responsible for a corresponding unimodular transformation of an $L$-system. Examples that illustrate the developed approach are presented.

The talk is based on joint work with K. A. Makarov and E. Tsekanovskiĭ (see references below).

[1] S. Belyi, K. A. Makarov, E. Tsekanovskiĭ,

[2] S. Belyi, K. A. Makarov, E. Tsekanovskiĭ,

Abstract:

We study Dirac operators on an infinite quantum graph of a bent chain form which consists of identical rings connected at the touching points by δ-couplings. It is established that the negativity of the coupling parameter is the necessary and sufficient condition for the existence of eigenvalues of the Dirac operator. The investigation is based on the transfer-matrix approach. It allows one to reduce the problem to an algebraic task. δ-couplings was introduced by the operator extensions theory method.

Abstract:

We discusses the inverse problem of how much information on an operator can be determined, or detected from ‘measurements on the boundary’. Our focus is on non-selfadjoint operators and their detectable subspaces (these determine the part of the operator ‘visible’ from ‘boundary measurements’).

We show results in an abstract setting, where we consider the relation between the M- function (the abstract Dirichlet to Neumann map or the transfer matrix in system theory) and the resolvent bordered by projections onto the detectable subspaces. More specifically, we investigate questions of unique determination, reconstruction, analytic continuation and jumps across the essential spectrum.

The abstract results are illustrated by examples of Schrodinger operators, matrix- differential operators and, mostly, by multiplication operators perturbed by integral oper- ators (the Friedrichs model), where we use a result of Widom to show that the detectable subspace can be characterized in terms of an eigenspace of a Hankel-like operator.

Abstract:

Numerical approximation of the Koopman operator gained attention in recent years because of the wide range of applications that was enabled by a data driven approach to approximation of spectral objects of Koopman operator. There are many different approaches to this problem but Dynamic Mode Decomposition (DMD) has shown to be a powerful tool for the analysis of nonlinear autonomous systems. However, existing DMD theory deals with autonomous systems and periodic and quasi-periodic forcing of the system. In this paper we propose an extension of Koopman operator framework from usual autonomous system setting to general non-autonomous setting with applications on linear and nonlinear dynamical systems. We propose a data driven algorithm based on DMD algorithm to obtain a time dependent finite dimensional approximation of Koopman operator.

Abstract:

We consider the non-stationary shear flow between two parallel solid and thermoinsulated horizontal plates, with the upper one moving irrotationally. The fluid is compressible, micropolar, viscous and heat-conducting, as well as is in thermodynamical sense perfect and polytropic. We assume that, given a Cartesian coordinate system $x$, $y$ and $z$, solutions of corresponding problem are $x$-dependent only.

In this work we present the existence and uniqueness results for corresponding problem with non-homogeneous boundary data for velocity and homogeneous boundary data for microrotation and heat flux, under the additional assumption that the initial density and initial temperature are strictly positive.

This is the joint work with Loredana Simčić.

Abstract:

In this talk I will present some highlights of joint investigations with D. Z. Arov over the past 20 plus years
on direct and inverse problems for canonical systems of integral and differential systems and related applications. The talk will focus on the inverse monodromy problem for $m\times m$ canonical differential systems
$$
y_t^\prime(\lambda)=i\lambda y_t(\lambda)H(t)J
$$
on a finite interval $[0,d]$,
where $H(t)$ is a summable $m\times m$ matrix valued function on $[0,d]$ that is positive semi-definite a.e. and $J$ is an
$m\times m$ signature matrix.

The inverse problem
is to recover the Hamiltonian $H(t)$ of the differential system from the monodromy matrix, i.e., from the value of the matrizant (fundamental solution) of the system at the right hand end point $d$ of the interval. This problem does not have a unique solution unless extra constraints are imposed.
Some known results will be reviewed briefly. Special classes of monodromy matrices for which the solutions of the inverse monodromy problem may be parametrized by identifying the matrizant with the resolvent matrices of a class of bitangential extension problems will be discussed. The development makes extensive use of two classes of reproducing kernel Hilbert spaces of vector valued entire functions that originate in the work of Louis de Branges and the interplay between them. Some new subclasses of these spaces and their role in the inverse monodromy problem will also be
discussed if time permits.

Abstract:

I will discuss a coupling problem for entire functions which arises in inverse spectral theory for singular second order ordinary differential equations/two-dimensional first order systems and is also of relevance for the integration of certain nonlinear wave equations.

Abstract:

We consider the controlled heat equation on rectangular
region $[0,2\pi L]^d$ with $d\geq 1$ and $L>0$ and control function
acting on a subset $\omega\subset [0,2\pi L]^d$ of positive measure.
We show that such a system is null-controllable at time $T>0$, i. e.
there exists a control function such that the solution of the system
is driven to zero at time $T$, and we give an estimate of the control
cost in terms of the geometric parameter of the problem.

Abstract:

The Friedrichs (1958) theory of positive symmetric systems of
first order partial differential equations encompasses many standard
equations of mathematical physics, irrespective of their type. This
theory was recast in an abstract Hilbert space setting by Ern, Guermond
and Caplain (2007), and by Antonić and Burazin (2010). In this work we
make a further step, presenting a purely operator-theoretic description
of abstract Friedrichs systems, and proving that any pair of abstract
Friedrichs operators admits bijective extensions with a signed boundary
map. Moreover, we provide sufficient and necessary conditions for
existence of infinitely many such pairs of spaces, and by the universal
operator extension theory (Grubb, 1968) we get a complete identification
of all such pairs, which we illustrate on two concrete one-dimensional
examples. This is a joint work with Nenad Antonić and Alessandro
Michelangeli.

Abstract:

The usual expectation concerning the spectrum of a
periodic quantum graph is a family of bands, some absolutely
continuous, some degenerate, and gaps, typically infinitely many of
them. The aim of this talk is to show that the picture could be
different. This is illustrated using two simple examples, a chain of
loops and a rectangular lattice. In particular, we are going to show
that (i) a chain in a homogeneous magnetic field can have no
absolutely continuous spectrum at all, (ii) a chain in a linear
magnetic can have a spectrum of a fractal nature, and (iii) even
without any external field a lattice can have a finite number of
spectral gaps in analogy with Bethe-Sommerfeld behaviour of the
`usual' Schrödinger operators. The last two effects depend on the
number-theoretic properties of the model parameters.

Abstract:

Multi-qubit quantum channel is considered. A possible error is related to the creation of entangled state during the transmission. A way of the channel quality estimation is suggested. It is based on the approximation of the unitary transfer-matrix of the channel by a Kronecker (tensor) product. A few examples are considered.

Abstract:

Spectral asymptotics of second order differential operators of the form $d/d\mu d/dx$ on the real line are well known if $\mu$ is a self similar measure with compact support. We extend the results to some more general cases such as random fractal measures. Moreover, we give a representation of the eigenfunctions as generalized trigonometric functions. The results were obtained in collaboration with Peter Arzt.

Abstract:

Abstract: We will illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, using this factorization method, we will derive a general inequality and demonstrate how particular choices of the parameters contained in this inequality yield well-known inequalities, such as the classical Hardy and Rellich inequalities as special cases. Actually, other special cases yield additional and apparently less well-known inequalities.

We will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators.

This is based on joint work with Lance Littlejohn and Michael Pang.

Abstract:

Using the method of subordinacy [1], the Weyl-Kodaira theorem [2], and multiplicity results to I.S. Kac [3], we construct a systematic method of spectral analysis for singular Sturm-Liouville operators on the line. The procedure enables the location, spectral type and multiplicity of the spectrum to be identified at all points on the line. If time allows we will also consider some simple examples and applications in order to illustrate the process.

References:

[1] D.J. Gilbert, On subordinacy and spectral multiplicity for a class of singular differential operators, Proc.
R. Soc. Edinburgh 128A, pp. 549-584, 1998.

[2] N. Dunford and J.T. Schwarz, Linear Operators, Part II, Chapter III.5, Interscience, 1963.

[3] I. S. Kac, On the multiplicity of the spectrum of a second order differential operator and the associated
expansion, Isv. Akad. Nauk SSSR, Ser. Mat. 27 (1963), pp. 1081 - 1112 pp. (in Russian).

Abstract:

Oscillation theory is known for almost two centuries and was mainly introduced by Charles-François Sturm and Joseph Liouville. This Sturm-Liouville oscillation theory can be studied for continuous Hamiltonian systems and discrete Jacobi operators. Here we consider Jacobi operators with block entries in a von Neumann algebra with finite trace, namely in a II$_1$ setting, and develop a rotation calculus for the spectrum of this Jacobi operator.

This is joint work with Hermann Schulz-Baldes and Carlos Villegas-Blas.

Abstract:

The Numerical Tensor Calculus enables computations of high-dimensional objects,
e.g., multivariate grid functions. In this lecture we start from an example in quantum
chemistry involving six-dimensional kernel functions and explain the technique of
Numerical Tensor Calculus with particular emphasis on the convolution operation.
A second subject is the tensorisation technique which also applies to one-dimensional
grid functions and allows to perform the convolution with a cost which may be much cheaper
than the fast Fourier transform.

Abstract:

In this talk Dirac operators with singular interactions supported on surfaces are discussed. Such operators can be used in relativistic quantum mechanics as idealized models for Dirac operators with strongly localized potentials.
After establishing self-adjointness of these operators, I will discuss the structure of their discrete and essential spectra and their convergence in the nonrelativistic limit.

Abstract:

It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties as the heat equation with a fractional Laplacian. In particular, the weak solution of the non-cutoff homogenous Boltzmann equation with initial datum in $L^1_2(\mathbb{R}^3)\cap L\log L(\mathbb{R}^3)$, i.e., finite mass, energy and entropy, should immediately become Gevrey regular.
So far, the best available results show that the solution becomes $H^{\infty}$ regular for positive times.
Gevrey regularity is also known for weak solutions of the linearised Boltzmann equation, where one studies solutions close to a Maxwellian distribution, or under additional decay assumptions on the solutions.
The main problem for establishing Gevrey regularity is that, in order to use the coercivity results on the non-cutoff Boltzmann collision kernel, one has to bound a non-linear and non-local commutator of the Boltzmann kernel with certain sub-exponential weights.
We prove, under the sole assumption that the initial datum is in $L^1_2(\mathbb{R}^3)\cap L\log L(\mathbb{R}^3)$, i.e., finite mass, energy and entropy, that the weak solution of the homogenous Boltzmann becomes Gevrey regular for strictly positive times. The main ingredient in the proof is a new way of estimating the non-local and non-linear commutator.

Joint work with Jean-Marie Barbaroux, Tobias Ried, and Semjon Wugalter.

Abstract:

We present recent results on the stability of collocation-quadrature methods based on Chebyshev nodes and applied to a class of Cauchy singular integral equations with additional Mellin operators having fixed singularities. The necessity and sufficience of the stability conditions is proved by $C^*$-algebra techniques. For example, it turns out that the part of the approximating operator sequence, associated with the Mellin part of the original equation, is ``very close'' to the finite section of particular operators belonging to a $C^*$-algebra of Toeplitz operators. Finally, these stability results are checked by applying the mentioned methods to integral equations for the two-dimensional elasticity problems of a notched half-plane and of a crack at a circular cavity surface. In particular, we will see that numerical results published in the book on two-dimensional elasticity theory by A. I. Kalandiya for the second problem seem not to be correct.

Abstract:

We will deal with a generalization of the spectral theorem for normal operators on Hilbert spaces
to normal, definitizable operators $N$ on Krein spaces $\mathcal K$.
Definitizability means the existence of $p\in \mathbb C[x,y]$ such that
\[
[p(\frac{N+N^*}{2},\frac{N-N^*}{2i})x,x] \geq 0
\]
for all $x\in \mathcal K$.
We consider the ideal $I$ generated by all such $p\in \mathbb C[x,y]$ and
assume $I$ to be zero-dimensional, i.e.\ $\dim \mathbb C[x,y]/I <\infty$. Applying primary factorization from
ring theory we have a unique decomposition $I=Q_1\cap\dots\cap Q_m$, where $Q_j$ is a primary ideal contained in a unique
maximal ideal $Q_j\subseteq P_j := \{f\in \mathbb C[x,y] : f(a_j)=0\}$ for pairwise distinct $a_j \in \mathbb C^2$.
We consider $V_{\mathbb R}(I) := \{a_j: a_j \in \mathbb R^2\}$ as a subset of $\mathbb C$ and $V(I)\setminus \mathbb R^2$ as a subset of $\mathbb C^2$.
For $w\in V_{\mathbb R}(I)$ define the finite dimensional algebras $\mathcal A(w):=\mathbb C[x,y]/(P_j\cdot Q_j)$, where $j$ is such that $a_j=w$,
and for $a\in V(I)\setminus \mathbb R^2$ define $\mathcal B(a):=\mathbb C[x,y]/Q_j$, where $j$ is such that $a_j=a$.
Now we consider functions $\phi$ which are defined on $\sigma(N) \dot\cup (V(I)\setminus \mathbb R^2)$ such that
$\phi(z) \in \mathcal A(z)$ for $z\in V_{\mathbb R}(I) \cap \sigma(N)$, $\phi(a) \in \mathcal B(a)$ for $a\in V(I) \setminus \mathbb R^2$,
$\phi$ is complex valued for all other point, and such that
for all non-isolated $w \in V_{\mathbb R}(I) \cap \sigma(N)$ the functions $\phi$ has a finite
Taylor like approximation near $w$.
Provided with pointwise addition and multiplication and some sort of conjugation
these functions $\phi$ form a $*$-algebra $\mathcal F_N$, and we show the existence of a $*$-algebra homomorphism
from $\mathcal F_N$ into $\{N,N^*\}''\subseteq B(\mathcal K)$.

Abstract:

We consider a quantum waveguide (e.g. a long wire with small cross section. The waveguide is homogeneous, but twisted in a random way. We investigate the influence of the random twist an the spectral properties of the waveguide. We prove that the twist produces Lifshitz tails at the bottom of the spectrum, i.e. the spectral density decays exponentially rapidly near the ground state energy.
Ina first step we reduce the problem to a one dimensional random Schrödinger operator whose potential is
the square of of an alloy-type random potential. Consequently we also have to deal with squared random potentials.

Abstract:

We solve the local equivalence problem for second order (smooth or analytic)
ordinary differential equations. We do so by presenting a complete
convergent normal form for this class of ODEs. The normal form is optimal in
the sense that it is defined up to the automorphism group of the model (flat)
ODE $y''=0$. For a generic ODE, we also provide a unique normal form. By doing
so, we give a solution to a problem which remained unsolved since the work of
Arnold. The method can be immediately applied to important classes of second
order ODEs, in particular, the Painleve equations.

Abstract:

We investigate the bottoms of the spectra and essential spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices.

The talk is based on a joint work with N. Nicolussi.

Abstract:

Abstract:

We study the generation of cosine families by second order differential operators with general boundary conditions on $L^p(\mathbb{R}_+,\mathbb{C}^l)\times L^p([0,1],\mathbb{C}^m)$. The abstract results are used to show well-posedness of wave- and diffusion equations on networks/metric graphs.
This is joint work with Klaus-Jochen Engel.

Abstract:

Boundary triples or quasi boundary triples can be used to describe closed extensions of symmetric operators with the help of an abstract boundary operator.
In this talk I shall discuss how the Weyl function corresponding to the
quasi boundary triple and the abstract boundary operator can be used to obtain spectral enclosures for such closed extensions.
In particular, I shall present sufficient conditions for extensions being m-sectorial or having spectrum contained
in a parabolic-type region.
The results are illustrated with elliptic operators with non-self-adjoint boundary conditions and with Schrödinger operators with $\delta$-potentials supported on hypersurfaces with complex-valued coefficients.

This talk is based on joint work with Jussi Behrndt, Vladimir Lotoreichik and Jonathan Rohleder.

Abstract:

This talk deals with an optimal control problems of Bolza type for a class of parabolic equations. The aim is to present a new methodology − based on a spectral decomposition of the operator governing the evolution of the system. Its implementation is described for a particular problem which consists in finding the initial datum that minimises a particular cost functional and ensures that the final state lies within a prescribed distance to a given target. The method leads us to an explicit expression of the optimal final state in terms of the given problem data. The obtained expression, combined with standard optimal control arguments enable construction of a one-shot algorithm providing an approximate solution. Its efficiency is assessed through numerical experiments.
Application of the method to a distributed control problem will be discussed as well.

Abstract:

The finite section method (FSM) aims to approximate the inverse of an infinite matrix $A$ by the inverses of growing finite submatrices along the main diagonal of $A$. We show that this works, with arbitrary cut-off points, for the Fibonacci Hamiltonian $A$ on the axis as well as on the half axis.

This is joint work with Hagen Söding (Hamburg).

Abstract:

We will discuss generalizations of the Faber-Krahn inequality
for the $1^{\text{st}}$ eigenvalue of the Robin Laplacian
with a negative boundary parameter on a complement
$\Omega^{\rm ext} := \mathbb{R}^d\setminus\overline{\Omega}$ of a bounded set
$\Omega\subset\mathbb{R}^d$. Here our focus will be on three types of constraints: $|\Omega| = {\rm const}$, $|\partial\Omega| = {\rm const}$, or fixed averaged mean curvatures of $\partial\Omega$. We will present results for $\Omega$ being:

- a bounded, convex domain in $\mathbb{R}^d$, $d \ge 3$;
- a bounded domain in $\mathbb{R}^2$ with $N\in\mathbb{N}$ simply connected components;
- an open arc in $\mathbb{R}^2$ with two endpoints;

These results are obtained largely in collaboration with David Krejčiřík.

Abstract:

Infinite quantum graphs with $\delta$-interactions at vertices are studied without any
assumptions on the lengths of edges of the underlying metric graphs. A connection
between spectral properties of a quantum graph and a certain discrete Laplacian given on
a graph with infinitely many vertices and edges is established. In particular, it is
shown that these operators are self-adjoint, lower semibounded, nonnegative, discrete,
etc. only simultaneously.

Spectral properties (e.g., selfadjointness, semiboundedness from
below, discreteness property, spectral types, etc.) of the quantum graph with
$\delta$-interactions will be discussed.

The talk is based on joint results with with P. Exner, A. S. Kostenko, H. Neidhardt
announced in [1].

[1] P. Exner, A. S. Kostenko, M. Malamud, and H. Neidhardt,

Abstract:

We introduce the essential numerical range $W_e(T)$ for closable Hilbert space operators $T$ and study its properties including possible equivalent characterizations and perturbation results. Although many of the properties known for the bounded case do not carry over to the unbounded case, the essential numerical range $W_e(T)$ allows us to describe the set of spectral pollution in a unified way when approximating $T$ by the Galerkin method or domain truncation method. If time permits we shall discuss some preliminary results for operator pencils.

Abstract:

We consider the symmetric differential system
\[
J y'-A(t)y=\lambda \Delta (t) y
\]
with $n\times n$-matrix coefficients $J(=-J^*=-J^{-1})$ and $A(t)=A^*(t), \; \Delta (t)\geq 0$ defined on an interval $[a,b) $ with the regular endpoint $a$. Let $\varphi (\cdot,\lambda)$ be a matrix solution of this system of an arbitrary dimension and let
\begin{equation*}
(Vf)(s)=\widehat f(s):=\int\limits_\mathcal I \varphi^*(t,s)\Delta(t)f(t)\,dt
\end{equation*}
be the Fourier transform of the function $f(\cdot)\in L_\Delta^2(\mathcal I)$. A pseudospectral function of the system is defined as a matrix-valued distribution function $\sigma(\cdot)$ of the dimension $n_\sigma$ such that $V$ is a partial isometry from $L_\Delta^2(\mathcal I)$ to $L^2(\sigma;\mathbb C^{n_\sigma})$ with the minimally possible kernel.

It is assumed that the deficiency indices $N_\pm$ of the system satisfies $N_-\leq N_+ = n$. For this case we define the monodromy matrix $B(\lambda)$ as a singular boundary value of a fundamental matrix solution $Y(t,\lambda)$ at the endpoint $b$ and parameterize all pseudospectral functions $\sigma(\cdot)$ of any possible dimension $n_\sigma\leq n$ by means of the
the linear-fractional transform
\begin{equation*}
m_\tau(\lambda)=(C_0(\lambda)w_{11}(\lambda)+C_1(\lambda)w_{21}(\lambda))^{-1}
(C_0(\lambda)w_{12}(\lambda)+C_1(\lambda)w_{22}(\lambda))
\end{equation*}
and the Stieltjes inversion formula for $m_\tau(\lambda)$. Here $w_{ij}(\lambda)$ are the matrix coefficients defined in terms of $B(\lambda)$ and $\tau=\{C_0(\lambda), C_1(\lambda)\}$ is a Nevanlinna matrix pair (a boundary parameter) satisfying certain admissibility conditions. It turns out that the matrix $W(\lambda)=(w_{ij}(\lambda))_{i,j=1}^2$ has the properties similar to those of the resolvent matrix in the extension theory of symmetric operators.

The obtained results develop the results by Arov and Dym; A. Sakhnovich, L. Sakhnovich and Roitberg; Langer and Textorius.

The results of the talk are partially specified in [1], [2]

1. V.I.Mogilevskii,

2. V.I.Mogilevskii,

Abstract:

We consider a block operator matrix of the form
$$
L=\left(\begin{array}{cc}
A_0 & B \\
B^* & A_1
\end{array}
\right),
$$
where $A_0$ and $A_1$ are self-adjoint operators in Hilbert spaces
$\mathfrak{H}_0$ and $\mathfrak{H}_1$, respectively, and $B$ is a
bounded operator from $\mathfrak{H}_1$ to $\mathfrak{H}_0$. It is
assumed that the operator matrix $L$ is in the Feshbach case, that
is, the spectrum of the entry $A_0$ overlaps the absolutely
continuous spectrum of the entry $A_1$. In addition, it is supposed
that $B$ is such that the Schur complement
$M(z)=A_0-z+B(A_1-z)^{-1}B^*$ admits analytic continuation in $z$ to
certain domains adjacent to the absolutely continuous spectrum of
$A_1$ in the so-called unphysical sheets of the spectral parameter plane
and, moreover, that the continued operator-function $M(z)$ admits
factorization in the sense of Markus and Matsaev (see [1]). The
corresponding operator roots of $M(z)$ allow us to construct
solutions to the operator Riccati equations associated with the block
operator matrices $L'$ emerging from $L$ as a result of the
so-called complex rotation of the entry $A_1$. In particular, the
complex scaling of $A_1$ is allowed if $A_1=-\Delta+V$
where $\Delta$ is the Lapalacian in $\mathbb{R}^n$ and $V$ an appropriate
analytic potential. The graph representation of invariant subspaces
of $L'$ associated with resonance (sub)sets of $L$ is discussed.

[1]
R. Mennicken and A. K. Motovilov,

Abstract:

Quantum graphs are collections of intervals glued at their endpoints in a network-like fashion, along with differential operators acting upon them. It is natural to consider elliptic equations associated with these structures: while the eigenvalues of quantum graph Laplacians may in principle be found as the zeros of a (trascendental) secular equation, this task is hard to pursue even for very simple quantum graphs, like stars. Since a pioneering paper by Nicaise in 1987, much attention has been devoted to derive a-priori spectral estimates that only depend on global quantities of combinatorial (like edge connectivity, total edge or vertex number), metric (like total length or diameter) or hybrid (like girth or the Cheeger constant) radius. We well review some recent advances in this field, especially those based on simple surgery principle that allow for spectral comparison of two different quantum graphs. Some of these ideas will also be extended to investigate changes of heat kernels upon graph operations.

This is joint work with Gregory Berkolaiko, James Kennedy and Pavel Kurasov.

Abstract:

We consider a coupled system, where the coupling occurs not via material properties but through an interaction on an interface separating the two regimes. Evolutionary well-posedness in the sense of Hadamard well-posedness supplemented by causal dependence is shown for a natural choice of generalized interface conditions. The results are obtained in a Hilbert space setting incurring no regularity constraints on the boundary and the interface of the underlying regions.

Abstract:

Joint work with B. Prangoski (University of Skopje) and J. Vindas (University of Gent) }

I will present our joint results on the Weyl
asymptotic formulae for the operators that are not of power-log-type as in the finite order (distributional) setting, but of
log-type, which in turn yields that the eigenvalues of infinite order $\Psi$DOs, with appropriate
assumptions, are "very sparse". As a by-product of our analysis, we also
obtain Weyl asymptotic formulae for a class of finite order Shubin $\Psi$DOs with some
conditions on the symbols that are not the ones usually discussed in the literature.
Moreover, I will present
infinite order Sobolev type spaces $H^*_{A_p,\rho}(f)$, where $A_p$ stend for Gevrey type sequences, while $\rho>0.$ $H^*_{A_p,\rho}(f)$ satisfies most of the familiar results for the classical, finite order, Sobolev spaces

Abstract:

Resonance states completeness problem for quantum graphs of various geometry having two semi-infinite edges is considered. Delta-coupling is assumed at the vertices. The problem is reduced to the scalar factorization problem in Sz.-Nagy functional model. Scattering problem is considered in the framework of Lax-Phillips approach.

Abstract:

Star-like quantum graph with delta-coupling at the vertex is considered. It is assumed that the lengths of edges vary in time. We deal with three different cases: Schrodinger equation on edges, wave equation on edges and Dirac equation on edges. Wave dynamics for these cases are compared. It is also made for another graph – a loop with attached segment. Dependence of the dynamics of the model parameters (delta-coupling parameter, type of length variation in time) is investigated.

Abstract:

We prove uniqueness in inverse acoustic scattering in the case the density of
the medium has an unbounded gradient across $\Sigma\subseteq\Gamma
=\partial\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{3}$
with a Lipschitz boundary. Such a result follows from a uniqueness result in
inverse scattering for Schrödinger operators with singular
$\delta$-type potential supported on the surface $\Gamma$ and of strength
$\alpha\in L^{p}(\Gamma)$, $p>2$.

Abstract:

We show a norm convergence result for the Laplacian on a class of post-critical fractals approximated by a sequence of finite-dimensional graph Laplacians. Similar arguments can also be used to approximate other Laplacians on suitable metric spaces by Laplacians on discrete weighted graphs.

As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm (joint work with Jan Simmer, University of Trier).

Abstract:

Inverses of structured matrices, namely Toeplitz, Hankel, and Toeplitz-plus-Hankel matrices, can be suitable
introduced as polynomials of two variables, called Bezoutians. How to obtain the coefficients of the involved
polynomials and matrix representations of the inverses is discussed.
Moreover, recent results of joint papers with Torsten Ehrhardt concerning the reverse problem - the inversion
of Bezoutians - are presented.

Abstract:

This is a joint work with Irada Dzhalladova.
In the contribution is investigated the navigation problem of following a moving target, using a mathematical model described by a system of differential equations with random parameters in the form
\begin{align}
\begin{split}
\dot{x}_1(t)&=\quad\quad\qquad\qquad\,\,-x_2(t)+b(\xi(t)),\\
\dot{x}_2(t)&=\alpha x_1(t)-(\delta+\beta)x_2(t)+\beta b(\xi(t)),
\end{split}
\end{align}
where $\xi(t)$ is a Markov process with two possible states.

The differential equations, which employ controls for following the target, are solved by a new approach using moment equations. The origin of the theory of moment equations and their use in the examination of the properties of solutions can be found in the works by K. Valeev and his scientific school, see, for instance, [5]. The moment equations method was used also in works [1] - [4] for studying stability of solutions to various kinds of systems with random structure.

The main result of the contribution provides us with a guarantee that a solution to derived system of ordinary linear differential equations - the moment equations, behaves as the mean value of a solution to the stochastic model of pursuit of a target. Therefore, instead of a solution to a complicated system of differential equations with random parameters we can solve the system of ordinary differential equations by known methods.

Simulations are presented to test effectiveness of the approach.

[1]
J. Diblík, I. Dzhalladova, M. Michalková, M. Růžičková,

[2]
J. Diblík, I. Dzhalladova, M. Michalková, M. Růžičková,

[3]
I. Dzhalladova, M. Růžičková, V. Štoudková Růžičková,

[4]
M. Růžičková, I. Dzhalladova,

[5]
K. G. Valeev, I. Dzhalladova,

Abstract:

In this talk Sturm-Liouville operators associated to singular indefinite differential expressions of the form
\[
\mathrm{sgn}(x) \left(-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+q(x)\right),\quad x\in\mathbb R,
\]
with a potential $q\in L^1(\mathbb R)$ are considered. Due to the weight $\mathrm{sgn}$, a self-adjoint realisation in the Hilbert space $L^2(\mathbb R)$ does not exist but it exist in a Krein space. Such an operator may have non-real spectrum. In the case $q\in L^1(\mathbb R)$ its essential spectrum is real. Hence, the non-real spectrum consists only of isolated eigenvalues. By adapting techniques of mathematical physics (Birman-Schwinger principle and WKB method) to the related eigenvalue problem we obtain three different bounds for the non-real point spectrum. These bounds depend only on the potential $q$.
This talk is based on a joint work with Jussi Behrndt (TU Graz) and Carsten Trunk (TU Ilmenau).

Abstract:

In [1], a Glivenko-Cantelli theory for almost additive functions on
lattices was developed, which allows to perform uniform thermodynamic
limits of monotone quantities like eigenvalue counting functions.
We present a vast generalization of the results to finitely generated
amenable groups replacing lattices. The main tools are Glivenko-Cantelli
theory and Ornstein-Weiss quasi-tilings, which substitute for tilings.
In the proof, we need to join the worlds of probability and geometric
group theory.

All results are joint work [2] with F. Schwarzenberger and I. Veselić.

[1] C. Schumacher, F. Schwarzenberger and I. Veselić. A Glivenko-Cantelli theorem for almost additive unctions on lattices, Stoch. Proc. Appl., 127 (1) 179-208, 2017

[2] C. Schumacher, F. Schwarzenberger and I. Veselić. Glivenko-Cantelli Theory, Ornstein-Weiss Quasi-Tilings, and Uniform Ergodic Theorems for Distribution-Valued Fields over Amenable Groups, preprint, submitted, 2017

Abstract:

We present a Glivenko Cantelli Theorem for a class of monotone, almost additive functions defined on lattices. While classical Glivenko Cantelli results deal with independent identical distributed random variables, we allow dependencies, associated to the underlying geometry of the lattice.
Our result not only show convergence, but also give quantitative estimates on the speed of convergence.
One main application we have in mind is the convergence of the eigenvalue counting functions for the lattice. Here our result shows the uniform existence of the associated limit function, the integrated density of states.
The presented results are published in a joint work with C. Schumacher and I. Veselić [1].
In a subsequent work [2] (also presented on the conference) we generalize this approach to amenable groups.

[1] C. Schumacher, F. Schwarzenberger and I. Veselić. A Glivenko-Cantelli theorem for almost additive functions on lattices, Stoch. Proc. Appl., 127 (1) 179-208, 2017

[2] C. Schumacher, F. Schwarzenberger and I. Veselić. Glivenko-Cantelli Theory, Ornstein-Weiss Quasi-Tilings, and Uniform Ergodic Theorems for Distribution-Valued Fields over Amenable Groups, preprint, submitted, 2017

Abstract:

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 π/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

Abstract:

We study the essential spectrum of so-called band-dominated operators on $L_p(G,\mu)$, where $G$ is a group, $\mu$ is a suitable measure on $G$ and $p\in (1,\infty)$. Such operators are norm-limits of band-operators, i.e. operators of finite range interaction. We characterize the Fredholm property by invertibility of the limit operators and obtain norm estimates of the essential norm of an operator on terms of the norm of its limit operators.
The results generalize previous ones for operators on $\ell_p(\mathbb{Z}^N)$ to an abstract setting which also includes the study of Toeplitz operators in Bergman-like spaces on $\mathbb{C}^d$ w.r.t. a gaussian measure.

This is joint work with Raffael Hagger (Leibniz Universität Hannover, Germany).

Abstract:

On the metric graph $\Gamma$ consider the Schrödinger operator $L_q = -\Delta + q$ with some electric potential $q \in L_1(\Gamma)$.
An upper estimate for the spectrum was obtained by Berkolaiko, Kennedy, Kurasov and Mugnolo in the case $q \equiv 0$.
We show that this estimate is in fact sharp by providing an explicit family of graphs and give a clear explanation by studying the related eigenfunctions.
In the second part we present a generalization of the inequality in the case $q \not\equiv 0$.

The talk is based on joint work with Pavel Kurasov, Stockholm University.

Abstract:

This is a joint work with Peter Šepitka. The comparative index was introduced by J. Elyseeva (2007) as an efficient tool in matrix analysis, which has fundamental applications in the discrete oscillation theory. In this talk we discuss the implementation of the comparative index into the theory of continuous time linear Hamiltonian systems (including Sturm-Liouville differential equations of even order). As a result we obtain new Sturmian separation theorems as well as new and optimal estimates for left and right proper focal points of conjoined bases of these systems. We derive our results for general possibly abnormal (or uncontrollable) linear Hamiltonian systems. The results turn out to be new even in the case of completely controllable systems.

Abstract:

First, we show a sufficient condition on the symbol of a (possibly non-self-adjoint) Toeplitz matrix guaranteeing the spectrum of its principal submatrix of an arbitrary size is purely real. In case of banded Toeplitz matrices, we provide a complete characterization of the class of matrices with the above property. Second, we describe how the eigenvalues of the principal submatrices distribute on the real line as the size goes to infinity. The presented results will be demonstrated by concrete examples.

Abstract:

In this talk we discuss a well-known result by Azizov and Iohvidov regarding the completeness and basis properties of the root vector system of a completely continuous operator in Pontryagin space [1]. We also cover the applications of this result to the studies of completeness and basis properties of the root vector system of some operator-valued functions [2].

Consider a class of quadratic operator pencils $L$ $$ L(\lambda)=\lambda^2 A+\lambda B+C, $$ where $A$, $B$, $C$ are continuous self-adjoint operators acting on a Hilbert space $H$. Here $A$ is a completely continuous operator, $B=B_1+B_2$, where $B_1$ is uniformly positive, $B_2$ is a compact operator, and the spectrum of the operator $-B_1^{-1}C$ has no more than countable set of accumulation points. Moreover, the set of regular points of the pencil $L$ is non-empty. The criterion of double completeness in $H$ of the system of root vectors of $L$ is obtained for this class of pencils in terms of non-degeneracy of root subspaces and their linearizers. It turns out that the system of root vectors for such pencils is twice complete and twice basisness in $H$ simultaneously. In the case of double completeness of the system of root vectors in $H$, there exists an almost $[\cdot,\cdot]$-orthonormal Riesz basis in $H$ that is constructed using Jordan chain vectors of the pencil $L$.

[1] T.Ya. Azizov, I.S. Iohvidov. Mathematical studies, 6, Proceedings of Academy of Science of Moldova, Kishinev, 1971.

[2] T.Ya. Azizov, I.S. Iohvidov. Indefinite Inner Product Spaces, Science, Moscow, 1996.

Abstract:

It has been shown that if the spectrum of the Laplacian with standard conditions on a metric graph $\Gamma$ coincides with the spectrum of the Neumann-Neumann Laplacian on a finite interval, then $\Gamma$ is in fact just the interval. This can be seen as a geometric version of Ambartsumian's classical Theorem. We will present a generalization of this result to Schrödinger operators with electric potential $q \in L_1(\Gamma)$, and present examples that show that this result can not be generalized to generic vertex conditions. This is joint work with J. Boman and P. Kurasov.

Abstract:

In this talk we are dealing with oscillation properties of conjoined bases of linear
Hamiltonian differential systems by employing the concept of the comparative index.
The comparative index was introduced by J. Elyseeva as an efficient tool in matrix
analysis, which has fundamental applications in the discrete oscillation theory.
Recently the author jointly with R. Šimon Hilscher and independently J. Elyseeva
implemented the comparative index into the theory of continuous time linear Hamiltonian
systems. We derived new and optimal estimates for the numbers of left and right proper
focal points of conjoined bases of these systems on bounded intervals. In this talk
we show that for a given nondegenerate compact interval there exist conjoined bases
with any number of left proper focals points and any number of right proper focal
points in the range between explicitly given minimal and maximal values. We also
present a construction of such a conjoined basis in terms of initial conditions
at a given point.

Abstract:

We study the Laplace operator on the 11 existing Archimedean lattices, i.e. graphs, based on vertex-transitive tilings of the plane by regular polygons.
It plays a role in understanding properties of materials such as graphene (Hexagonal lattice) or certain jarosites (Kagome lattice).
Archimedean lattices may or may not exhibit eigenfunctions of finite support…
The occurrence of finitely supported eigenfunctions corresponds to jumps in the Integrated Density of States (IDS) and is related to exotic material behavior.

Using Floquet theory, we can rather explicitely calculate the IDS (as well as the Density of States) of all 13 Archimedean lattices and thus identify all occurring finitely supported eigenfunctions.

This is joint work with Norbert Peyerimhoff.

Abstract:

We study random Schrödinger operators in $L^2 (\mathbb{R}^d)$ of the type
\[
H_\omega = -\Delta + V_\omega
\]
where $\Delta$ denotes the Laplace operator and $V : \Omega \times \mathbb{R}^{d} \to \mathbb{R}$ is a stationary jointly measurable Gaussian field on a complete probability space $(\Omega , \mathcal{F} , \mathbb{P})$. If the covariance function decays exponentially (but may change its sign arbitrary) we prove a Wegner estimate for finite volume restrictions of $H_\omega$.

In order to treat such indefinite correlations, we prove the following tiling theorem. Let $f \in L^1 (\mathbb{R}^d)$ be exponentially decaying. Then there are $C_1 \not = 0$ and $I_0 \in \mathbb{N}_0^d$ such that
\[
\forall x \in \mathbb{R}^d \colon \quad \int_{\mathbb{R}^d} \frac{y^{I_0}}{C_1} f (x-y) \mathrm{d}y = 1 .
\]
This is a joint work with Ivan Veselić.

Abstract:

We consider a bounded connected open set
$\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite
$(d-1)$-dimensional Hausdorff measure. Then we define the
Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form
methods. The operator $-D_0$ is self-adjoint and generates a
contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on
$L_2(\Gamma)$. We show that the asymptotic behaviour of
$S_t$ as $t \to \infty$ is related to properties of the
trace of functions in $H^1(\Omega)$ which $\Omega$ may or
may not have. We also show that they are related to the
essential spectrum of the Dirichlet-to-Neumann operator.

The talk is based on a joint work with W. Arendt (Ulm).

Abstract:

The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.

Based on joint work with Tom Koornwinder and Aleksey Kostenko.

Abstract:

A prominent class of objects studied in ${\mathcal P}{\mathcal T}$ symmetric quantum mechanic
consists of the ${\mathcal P}{\mathcal T}$ symmetric Hamiltonians
\begin{equation*}
(\tau y) (x) := -y^{\prime\prime}(x) + x^2(ix)^\epsilon y(x), \quad \epsilon \geq 2,
\mbox{ and }x\in\Gamma,
\end{equation*}
where $\Gamma$ is a contour in $\mathbb C$ which is, in general, different from the real line
and satisfies, according to the rules of ${\mathcal P}{\mathcal T}$ symmetry, some additional conditions.

To the differential expression $\tau$ one can associate
a ${\mathcal P}{\mathcal T}$ symmetric operator which is simultaneously selfadjoint in the Krein space $(L_2(\mathbb
R),[\cdot,\cdot])$, where
$[\cdot,\cdot]$ is given by
$[\cdot,\cdot] :=
(\mathcal P\, \cdot,\cdot)$ and
$(\cdot,\cdot)$ stands for
the usual $L_2$-product, $\mathcal P$ is the parity.

We will show that the spectrum of such an operator
consists of isolated eigenvalues only which accumulate at $\infty$.
Moreover, we discuss the location of the (point) spectrum of such operators and
we will determine areas in the complex plane which are free of eigenvalues.
Contrary to physical intuition, we will single out many cases where the real
axis does only contain finitely many eigenvalues, i.e. there are infinitely many
eigenvalues in the non-real plane.

The talk is based on a joint work with Florian Leben (TU Ilmenau).

Abstract:

We consider magnetic Dirichlet Laplacian on a layer of a fixed width constructed along a two-dimensional hypersurface. The magnetic field will always be uniform, but the geometry of the layer may be non-trivial. If the layer is planar and the field is perpendicular to it the spectrum consists of infinitely degenerate eigenvalues. We will study translationally invariant geometric perturbations and derive several sufficient conditions under which the spectrum, in its entirety or a part of it, becomes absolutely continuous. The talk is based on a joint work with T. Kalvoda and P. Exner.

Abstract:

Spectral theory for general classes of first order systems has been less
popular since 1990's.
In this talk, I would like to propose an approach which can
deal with Maxwell and Dirac operators in a unified manner,
and introduce a new topic for these operators.
Namely, I will discuss space-time estimates
for these operators.

This talk is based on joint work with Matania Ben-Artzi (Hebrew University).

Abstract:

I shall present some recent work done in collaboration with V. Banica and F. De La Hoz about the evolution of vortex filaments according to so called binormal flow. I will exhibit a non-linear Talbot effect and some results concerning the transfer of energy and of linear momentum. The relation with Frisch-Parisi conjecture in turbulence will be also discussed.

Abstract:

We study functional differential and integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the considered equations is an abstract hyperbolic equation perturbed by terms with delay and terms containing Volterra integral operators.
We prove that the initial boundary-value problems in weighted Sobolev spaces are well posed on the positive semi-axis for the specified equations. We also research the spectra of operator-valued functions that are symbols of these equations in the autonomous case.
We study the spectral problems for operator-valued functions that are symbols of Volterra integrodifferential equations with unbounded operator coefficients in Hilbert spaces. Operator models of such type have many applications in the linear viscoelasticity theory, homogenization theory, heat conduction theory in media with memory, etc.

References

[1] N.A. Rautian and V.V. Vlasov, Properties of solutions of integro-differential equations arising in heat and mass transfer theory, Transactions of the American Mathematical Society. 2014, vol. 75, рр. 185-204.

Abstract:

Let $J$ be the Lévy density of a symmetric Lévy process in $\mathbb{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator
$$
\mathcal{L}^{\kappa}f(x):= \lim_{\varepsilon \downarrow 0} \int_{\{z \in \mathbb{R}^d: |z|>\varepsilon\}} (f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, ,
$$
where $\kappa(x,z)$ is a Borel function on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<\kappa_0\le \kappa(x,z)\le \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$ and$|\kappa(x,z)-\kappa(y,z)|\le \kappa_2|x-y|^{\beta}$ for some $\beta\in (0, 1]$.
Following the work of Z.-Q. Chen and X. Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups,

Joint work with P. Kim (SNU) and R. Song (UIUC)

Abstract:

We introduce an abstract framework for a maximally dissipative operator A and its anti-dissipative adjoint and use this framework to construct the selfadjoint dilation of A using the Straus characteristic function. The advantage of this construction is that the parameters arising in the dilation are explicitly given in terms of parameters of A (such as coefficients of a differential expression). Having constructed the selfadjoint dilation, we will discuss its spectral representation. The abstract theory will be illustrated by the example of dissipative Schrodinger operators.

Abstract:

The following theorem holds:
Let $\mathcal L$ be a - not necessarily nondegenerated or complete - positive semidefinite inner product space carrying
an anti-isometric involution, and let $S$ be a symmetric operator in $\mathcal L$.
If $S$ possesses a universal directing functional $\Phi:\mathcal L\times\mathbb C\to\mathbb C$ which
is real w.r.t. the given involution,
and the closure of $S$ in the completion of $\mathcal L$ has defect index $(1,1)$, then there exists a de Branges (Hilbert-) space
$\mathcal B$ such that $x\mapsto\Phi(x,\cdot)$ maps $\mathcal L$ isometrically onto a dense subspace of $\mathcal B$ and
the multiplication
operator in $\mathcal B$ is the closure of the image of $S$ under this map.

In this paper we consider a version of universal directing functionals defined on an open set $\Omega\subseteq\mathbb C$
instead of the whole plane, and inner product spaces $\mathcal L$ having finite negative index.
We seek for representations of $S$ in a class of reproducing kernel almost Pontryagin spaces of functions on $\Omega$
having de Branges-type properties.
Our main result is a version of the above stated theorem, which gives conditions making sure that $\Phi$ establishes such
a representation. This result is accompanied by a converse statement and some supplements.

As a corollary, we obtain that if a de Branges-type inner product space of analytic functions on $\Omega$
has a reproducing kernel almost Pontryagin space completion,
then this completion is a de Branges-type almost Pontryagin space.
This is an important fact in applications.
The corresponding result in the case that $\Omega=\mathbb C$ and $\mathcal L$ is positive semidefinite is well-known, often used,
and goes back (at least) to work of M.Riesz in the 1920's.

Abstract:

We present a general theory of rank 2 perturbations of matrices. Several special cases will be discussed. We shall also present the perturbatin of spectra. In particular the interlacing property and the limit behaviours will be discussed.
The talk is based on the paper:

A. Kula. M. Wojtylak, J. Wysoczanski, "Rank 2 perturbation of matrices and operators and operator model for t-transformation of probability measures", J. Funct.Anal. 272 (2017), no.3

Abstract:

We shall discuss the application of the theory of operators in problems related to noncommutative probability. In particular we shall discuss two special perturbations - the "diagonal" and the "antiiagonal" - which are related to some deformations of probability measures in free probability. The relation between operators and probability measures - their distributions - will be presented.

The talk is based on the paper:
A. Kula. M. Wojtylak, J. Wysoczanski, "Rank 2 perturbation of matrices and operators and operator model for t-transformation of probability measures", J. Funct.Anal. 272 (2017), no.3

Abstract:

Square integrable solutions and principal solutions play (separately) important roles in the qualitative theory of the linear Hamiltonian differential systems
\begin{equation*}\tag{$\text{H}_\lambda$}\label{Hla}
z'(t,\lambda)=[\mathcal{H}(t)+\lambda\,\mathcal{J}\,\mathcal{W}(t)]\,z(t,\lambda),
\quad \mathcal{J}:=\begin{pmatrix} 0 & I\\ -I & 0\end{pmatrix},
\end{equation*}
where $t\in[a,\infty)$, $\lambda\in\mathbb{C}$ is a spectral parameter, and $\mathcal{H}(t)$ and $\mathcal{W}(t)$ are piecewise continuous even order matrix-valued functions such that $\mathcal{J}\mathcal{H}(t)+\mathcal{H}^*(t)\mathcal{J}=0$ and $\mathcal{W}(t)=\mathcal{W}^*(t)\geq0$ for all $t\in[a,\infty)$. We present our recent results in the Weyl-Titchmarsh theory for system \eqref{Hla} which were derived by using principal solutions. In particular, we show a close connection between the Weyl
solution and the principal solution; compare with Theorems 2.13 and 3.11 in [1] for the second order Sturm-Liouville differential equations.
The talk is based on the joint research with R. Šimon Hilscher, see [2].

[1]
S. L. Clark, F. Gesztesy, and R. Nichols,

[2]
R. Šimon Hilscher and P. Zemánek,