IWOTA 2019

International Workshop on

Operator Theory and its Applications

IWOTA 2019

International Workshop on

Operator Theory and its Applications

July 22-26, 2019

Instituto Superior Técnico

Lisbon, Portugal

Instituto Superior Técnico

Lisbon, Portugal

- Nikita Artamonov

MGIMO University ## Solvability of an Operator Riccati Integral Equation in a Reflexive Banach Space

If $X_{1,2}$ are Banach spaces, then by $\mathcal{L}(X_1,X_2)$ we will denote the space of bounded operators acting from $X_1$ to $X_2$. By $C_s(\mathcal{I}; \mathcal{L}(X_1,X_2))$ we denote the space of strongly continuous operator functions on interval $\mathcal{I}=[0,T]$ with the topology of strongly uniform convergence.

An operator function $\{U_{t,s}\}_{0\leq s\leq t\leq T}$ on a Banach space is called

*forward (in time) evolution family*if $U_{t,t}=I$ and $U_{t,s}=U_{t,r}U_{r,s}$ for all $0\leq s\leq r\leq t\leq T$. An operator function $\{V_{s,t}\}_{0\leq s\leq t\leq T}$ on a Banach space is called*backward (in time) evolution family*if $V_{t,t}=I$ and $V_{s,t}=V_{s,r}V_{r,t}$ for all $0\leq s\leq r\leq t\leq T$. An evolution family is called*strongly continuous*if it is strongly continuous in $t$ (for fixed $s$) and in $s$ (for fixed $t$).Let $X$ be a reflexive Banach space with duality pairing $\langle f,x\rangle$ ($x\in X, f\in X^*$). If $A_1\in\mathcal{L}(X,X^*)$ then taking into account the canonical isomorphism between $X$ and $X^{**}$ one can consider that the adjoint operator $A_1^*\in\mathcal{L}(X,X^*)$. Operator $A_1\in\mathcal{L}(X,X^*)$ is self-adjoint if $A_1=A_1^*$. Self-adjoint $A_1\in\mathcal{L}(X,X^*)$ is non-negative if $\langle A_1x,x\rangle\geq0$ for all $x\in X$. Analogously if $A_2\in\mathcal{L}(X^*,X)$ then one can consider that the adjoint operator $A_2^*\in\mathcal{L}(X^*,X)$. Operator $A_2\in\mathcal{L}(X^*,X)$ is self-adjoint if $A_2=A_2^*$.

Self-adjoint operator $A_2\in\mathcal{L}(X^*,X)$ is non-negative if $\langle x,A_2x\rangle\geq 0$ for all $x\in X$. Note that if $U_{t,s}$ is strongly continuous evolution family in reflexive space $X$ then $V_{s,t}=U^*_{t,s}$ is strongly continuous backward evolution family in $X^*$.

**Theorem**.*Let $X$ be a reflexive Banach space and the following assumptions hold*:- $\{U_{t,s}\}_{0\leq s\leq t\leq T}$
*is strongly continuous and uniformly bounded forward evolution family in*$\mathcal{L}(X)$.*Then*$V_{s,t}=U^*_{t,s}$*is strongly continuous and uniformly bounded backward evolution family in*$\mathcal{L}(X^*)$. -
*operator functions*$C\in C_s(\mathcal{I};\mathcal{L}(X,X^*))$ and $B\in C_s(\mathcal{I};\mathcal{L}(X^*,X))$ - $C(t)=C^*(t)\geq0$ and $B(t)=B^*(t)\geq0$ for all $t\in\mathcal{I}$.

*Then for all self-adjoint non-negative*$G\in\mathcal{L}(X,X^*)$*the (backward) integral Riccati equation*\[P(t)=V_{t,T}GU_{T,t}+\int_t^TV_{t,s}\{C(s)-P(s)B(s)P(s)\}U_{s,t}ds \]*has a unique self-adjoint non-negative solution*$P\in C_s(\mathcal{I};\mathcal{L}(X,X^*))$.Some applications to the solvability of a system of forward-backward evolution linear equations \[ \begin{pmatrix} x'(t) \\ y'(t) \end{pmatrix}=\begin{pmatrix} A(t) & -B(t) \\ -C(t) & -A^*(t) \end{pmatrix} \begin{pmatrix} x(t) \\ y(t) \end{pmatrix},\quad \begin{aligned} x(0) &=x_0 \\ y(T) &=G x(T) \end{aligned}\quad t\in[0,T]\] and mean-field game system of PDE will be given.

### Reference

N. Artamonov,

*Solvability of an Operator Riccati Integral Equation in a Reflexive Banach Space*. Differential Equations, 2019, Vol. 55, No. 5, pp. 718–728- $\{U_{t,s}\}_{0\leq s\leq t\leq T}$
- Dmytro Baidiuk

Tampere University ## Robustness of polynomial stability for generalized wave equations

We consider a generalized damped wave equation with some strictly negative operator $L$ instead of the Laplacian operator. Such equation can be represented as the first order system, where operator $A$ generates the strongly continuous semigroup of contractions. It has been shown in the literature that a wide class of damped wave equations are only polynomially stable (instead of being uniformly exponentially stable). We investigate preservation of such stability in the case if the operator $A$ is perturbed with finite rank operators or Hilbert-Schmidt operators. It is well known that the robustness properties of the polynomial stability are weaker than the robustness properties of the exponential stability. In [Paunonen 2014] was shown that the polynomial stability is preserved if the graph norms of fractional powers of $A$ applied to perturbation operators are small enough. However, in general case the fractional powers of dissipative operators are difficult or even impossible to compute. In this presentation, we show that for generalized wave equation it is possible to reduce computations of fractional powers of the dissipative operator $A$ to computations of fractional powers of the strictly positive operator $(-L)$. That makes the conditions for robustness easier to verify.

We consider two applications of the perturbation results. In the first example we apply the results to a 2-D damped wave equation, but additionally we suppose that the damping function does not satisfy the Geometric Control Condition. For such example, it is shown that the polynomial stability is preserved if some vectors (connected to the perturbation operators) belong to fractional Sobolev spaces. Our second example is a Webster's equation with a weak damping.

This is joint work with Lassi Paunonen.

- Christian Budde

University of Wuppertal ## Bi-Continuous Flows for Flows in Networks

We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the $L^{\infty}$-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions. This talk is based on joint works with M. Kramar Fijavz.

- Jorge Gonzalez Camus

Universidad de Santiago de Chile ## Fundamental solutions for dynamical systems involving the fractional Laplacian

In this talk we give the representation results for solutions of a time-fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example chain processes in chemistry and radioactivity. Our focus is in the problem \[ \begin{cases} \mathbb D^\beta_t u(n,t)= -(-\Delta_d)^\alpha u(n,t) + g(n,t),\, &n\in\mathbb{Z}, \; t\gt 0,\\ u(n,0)=\varphi(n),\quad u_t(n,0)=\psi(n),\; &n\in\mathbb{Z},\\ \end{cases}\] where $0\lt \beta\leq2$, $0\lt \alpha\leq 1$, $n\in\mathbb Z$, $(-\Delta_d)^{\alpha}$ is the discrete fractional Laplacian and $\mathbb{D}_t^{\beta}$ is the Caputo fractional derivative of order $\beta$. We discuss important special cases as consequences of the representations obtained, such as the Heat and Wave semidiscrete equations.

- Tom ter Elst

University of Auckland, Department of Mathematics ## The Dirichlet-to-Neumann operator on $C^{1+\kappa}$-domains

We present some recent results on kernel bounds for the semigroup generated by the Dirichlet-to-Neumann operator when the underlying operator has Hölder continuous coefficients and the domain has a $C^{1+\kappa}$-boundary. The proof depends on Gaussian bounds for derivatives of the semigroup kernel of an elliptic operator with Dirichlet boundary conditions. As a consequence the Dirichlet-to-Neumann semigroup is holomorphic on the right half-plane on $L_1$.

This is joint work with El Maati Ouhabaz.

### References

- Elst, A. F. M. ter and Ouhabaz, E.-M.,
*Dirichlet-to-Neumann and elliptic operators on $C^{1+\kappa}$-domains: Poisson and Gaussian bounds*. J. Differential Equations (2019). https://doi.org/10.1016/j.jde.2019.04.034. - Elst, A. F. M. ter and Ouhabaz, E.-M.,
*Analyticity of the Dirichlet-to-Neumann semigroup on continuous functions*. J. Evol. Eq.**19**(2019), 21-31.

- Elst, A. F. M. ter and Ouhabaz, E.-M.,
- Kamal Khalil

Cadi Ayyad University, Faculty of Sciences Semlalia ## Compact almost automorphic solutions to semilinear parabolic evolution equations

The aim of this work is to study the existence of compact almost automorphic solutions to semilinear evolution equations in Banach spaces. We assume that the linear part is the infinitesimal generator of a compact $C_0$-semigroup and the nonlinearity satisfying the weakest condition that is almost automorphic in Stepanov sense in the first variable and just continuous in the second one. Using the subvariant functional method introduced by Fink [1], we give sufficient condition ensuring the existence of a compact almost automorphic solution on $\mathbb{R}$ provided the existence of at least one bounded solution on $\mathbb{R}^+$. For illustration, we propose to study a class of nonautonomous reaction-diffusion problems. We introduce the fact that for given unbounded forcing term of the problem, we have a unique bounded global solution on the right half-line.

Joint work with Brahim Es-sebbar and Khalil Ezzinbi.

### Reference

- A. M. Fink, Almost automorphic and almost periodic solutions which minimize functional, Tôhoku Math. J., 20.2, 323-332, (1968).

- Karsten Kruse

Hamburg University of Technology ## Subordination for bi-continuous semigroups

In this talk we consider uniformly bounded (locally) bi-continuous semigroups $(T_t)$ on $X$ w.r.t. $\tau$ with generator $-A$ where $(X,\|\cdot\|)$ is a Banach space and $\tau$ a coarser Hausdorff locally convex topology on $X$.

For every Bernstein function $f$ we show that the sequential closure of $-f(A)$ generates a uniformly bounded (locally) bi-continuous semigroup on $X$ if $X$ is a sequential space once it is equipped with the topology obtained from mixing $\tau$ and the norm topology. This new semigroup which is of the same "kind" as $(T_t)$ is called the subordinated semigroup to $(T_t)$ w.r.t. $f$. As a special case we obtain that the sequential closures of fractional powers $-A^{\alpha}$, where $\alpha \in (0,1)$, are generators.

This is a joint work with Jan Meichsner and Christian Seifert.

- Wha-suck Lee

University of Pretoria, South Africa ## Evolution operators in empathy with a convolution semigroup

In probability theory, Fokker-Planck equations are the partial differential equations governing the transition function of the Markov process. In the evolution equations approach, partial differential equations are rewritten as ordinary differential equations in Banach spaces. For the first time in

**[2]**, the Fokker Planck equation for a pair of discontinuous intertwined Markov processes was shown to have the form of an implicit evolution equation, \begin{equation}\frac{d}{dt}[Bu(t)] = Au(t),\end{equation} where the symbols $A$ and $B$ denote unbounded linear operators with a common domain*$D$*in a Banach space $X$ and range in a*distinct*Banach space $Y$.In this talk we consider the continuous analogue in the form of two homogeneous Markov transition functions intertwined by the extended ChapmanKolmogorov equation. Abstract harmonic analysis techniques are used to (i) construct a noncommutative extension of the Feller convolution needed to handle the non-commutative dynamics of the extended Chapman-Kolmogorov equation and (ii) extend the Riesz representation to two-state space distributions as admissible homomorphisms introduced in

**[1]**.### References

- W.-S. Lee and N. Sauer, Intertwined evolution operators,
*Semigroup**Forum***94**(2017), 204–228. - W.-S. Lee and N. Sauer, Intertwined Markov processes: the extended Chapman-Kolmogorov equation,
*Proc. R. Soc. Edinb. A***148**(2018), 123–131.

- W.-S. Lee and N. Sauer, Intertwined evolution operators,
- Jesús Oliva Maza

Universidad de Zaragoza ## Subordination of the generalized Stieltjes operator in terms of $C_0$-groups

In this talk, we present our recent work concerning the the generalized Stieltjes operator $$\mathcal{S}_{\beta,\mu} f(t):={t^{\mu-\beta}}\int_0^\infty{s^{\beta-1}\over (s+t)^{\mu}}f(s)ds, \qquad t\gt 0, $$ defined on Sobolev spaces $\mathcal{T}_p^{(\alpha)}(t^\alpha)$ (where $\alpha\ge 0$ is the fractional order of derivation and are embedded in $L^p(\mathbb{R}^+)$). By subordinating these operators in terms of $C_0$-groups and transfering new properties of some special functions, we are able to prove that they are bounded operators when the parameters $\mu, \beta$ are properly chosen. Under this conditions, it is possible to calculate and represent explicitly their spectrum set $\sigma (\mathcal{S}_{\beta,\mu})$ and their operator norms (which depend on $p$). Even more, it can be shown that they commute and factorize with generalized Cesáro operator on $\mathcal{T}_p^{(\alpha)}(t^\alpha)$. This is joint work with Pedro J. Miana.

- Jan Meichsner

Technische Universität Hamburg, Institut für Mathematik, Lehrstuhl für angewandte Analysis ## Description of fractional powers via Caferelli-Silvestre extension

In 2007 L. Caffarelli and L. Silvestre attracted a lot of attention when describing the action of the fractional Laplacian $-\Delta^{\alpha}$ on a function $f$ in some function space using a solution $u$ of a second order ODE with singular coefficients, which is essentially Bessel’s differential equation, in the aforementioned space. One may generalise to arbitrary sectorial operators $A$ instead of $-\Delta$ and consider the ODE in some general Banach space $X$. Until now many other authors contributed to this more general framework. The talk aims to answer the questions whether the considered ODE always has a unique solution and to what extend this solution can be used to describe fractional powers of $A$.

This is joint work with Christian Seifert.

- Marco Peruzzetto

Kiel University ## On Eventual Regularity Properties of Operator Valued Functions

For two Banach spaces $X,Y$ let $u:\mathbb{R}_{\geq 0}\rightarrow \mathcal{L}(X;Y)$ be an operator valued function and $\mathtt{P}$ a regularity property. Assume that each orbit $t\mapsto u(t)x$ has the regularity property $\mathtt{P}$ on some interval $(t_x,\infty)$ in general depending on $x\in X$. In this context we prove a Baire-type theorem, which allows to remove the dependency of $x$ in certain situations. Afterwards, we provide some applications which are of interest in semigroup theory. In particular, we generalize and explain the result obtained by T. Bárta in his article

*Two notes on eventually differentiable families of operators*(Comment. Math. Univ. Carolin. 51,1 (2010), 19-24).- Piotr Polak

University of Szczecin ## Decay/growth rate estimation for certain class of strongly continuous semigroups

We consider linear differential equation with operator generating a strongly continuous semigroup in Banach space. The talk is devoted to several aspects of stability and asymptotic behaviour of the semigroups and the corresponding solutions. We discuss certain spectral conditions of asymptotic stability and present the generalizations of classical stability concept: polynomial stability and the existence of the fastest growing solution - so called maximal asymtotics. In particular, we give a conditions for polynomial stability in terms of location of the spectrum of generator and their rate of approaching to imaginary axis.

- Felix Schwenninger

University of Twente, The Netherlands ## Around Baillon’s theorem on maximal regularity

Baillon’s theorem asserts the following dichotomy for a semigroup generator $A$ on a Banach space $X$: If $A$ has the maximal regularity property with respect to the space of continuous functions, then $A$ is either bounded or $X$ contains $c_{0}$. A corresponding statement holds for $L^{\infty}$ functions. We will discuss a variant of such maximal regularity notions and its consequences in spirit of Baillon’s result.

- David Seifert

University of Oxford ## Energy decay of damped waves

Consider the classical

*damped wave equation*$$u_{tt}(x,t)+b(x)u_t(x,t)-\Delta u(x,t)=0,\quad x\in\Omega,\ t\gt 0,$$ to be solved subject to the Dirichlet boundary condition $u=0$ on $\partial\Omega$ for a suitable initial displacement and velocity. Here $\Omega$ is a bounded domain in $\mathbb{R}^n$ for some $n\ge1$ (or more generally a compact Riemannian manifold) and $b\in L^\infty(\Omega)$ is a given non-negative function. The damped wave equation is well posed and gives rise to a $C_0$-semigroup $(T(t))_{t\ge 0}$ of contractions on the Hilbert space $H_0^1(\Omega)\times L^2(\Omega)$. The natural inner product on this space is such that the (squared) norm of any semigroup orbit corresponds to the*energy*$$E_u(t)=\frac12\int_\Omega |\nabla u(x,t)|^2+|u_t(x,t)|^2\,\mathrm{d}x,\quad t\ge 0,$$ of the associated mild solution $u$ of the damped wave equation. It is relatively straightforward to see that as soon as the damping coefficient $b$ is non-trivial the energy of any solution $u$ satisfies $E_u(t)\to0$ as $t\to\infty$. On the other hand, the question*how fast*the energy decays is a difficult one and has received a considerable amount of attention over the past three decades.In a series of deep results from the 1990s it was shown by Rauch, Taylor, Bardos, Lebeau, Burq and Gérard that for continuous damping coefficients $b$ the semigroup $(T(t))_{t\ge0}$ is uniformly exponentially stable if and only if the domain of damping $\{x\in\Omega:b(x)>0\}$ satisfies the so-called

*geometric control condition*. Ever since, attention has turned primarily to obtaining rates of energy decay for sufficiently regular solutions in the case where the geometric control condition is violated. The two central challenges in the modern*frequency-domain approach*are, first, to obtain good estimates for the rate at which the resolvent norms $\|(is-A)^{-1}\|$ of the semigroup generator $A$ grow as $|s|\to\infty$ and, second, to deduce precise decay estimates for suitable semigroup orbits. The former typically requires one to exploit the specific features of the particular problem at hand (often using delicate arguments from microlocal analysis), while the latter can be approached using elegant general techniques from complex and harmonic analysis.In this talk I will give a brief overview of some of the main results in the modern literature on quantified energy decay of damped waves, including in particular some recent results obtained in joint work with J. Rozendaal (Canberra and Warsaw) and R. Stahn (formerly TU Dresden), which permit lossless conversion of resolvent estimates into decay rates for the semigroup under what are essentially the mildest possible conditions.

- Christian Seifert

Hamburg University of Technology, Germany ## On Observability Estimates in Banach Spaces

In this talk we study sufficient conditions for observability of systems in Banach spaces.

In an abstract Banach space setting we show that an uncertainty relation together with a dissipation estimate implies an observability estimate with explicit dependence on the model parameters. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider elliptic operators on $L_p$ spaces. Combined with the well known relation between observability and controllability we derive sufficient conditions for null-controllability and bounds on the control cost.

The talk is based on joint work with Dennis Gallaun and Martin Tautenhahn.

- Grigory Sklyar

University of Szczecin, Institute of Mathematics ## Hardy Inequality in Estimation of Semigroups Growth

In the talk we discuss behavior of $C_0$ groups generated by infinitesimal operators with discrete spectrum and eigenspaces of uniformly bounded dimension. In the recent works of Xu & Yung and Zwart it is shown that if in addition the eigenvalues are separated that corresponding eigenspaces form a genaralized Riesz basis. In 2017 we showed that the condition of separatness is necessary for Riesz basis property. More specifically we give examples of semigroups which generator has the spectrum consisting of pure imagionary eigenvalues $\lambda_n=i\log n$ but corresponding eigenvectors do not form even Schauder basis. Further in the talk we show that in this case corresponding $C_0$ group $T(t)$ may increase as arbitrary power of $t$.

- El Gantouh Yassine

IBN ZOHR, Faculty of Sciences Agadir ## Controllability of Network Systems

In this presentation, the rich feedback theory of regular linear systems in the Salamon-Weiss sense as well as some advanced tools in semigroup theory, and graph theory are used to investigate the relative controllability associated with network systems.