IWOTA 2019

International Workshop on
Operator Theory and its Applications

July 22-26, 2019
Instituto Superior Técnico
Lisbon, Portugal

# Abstracts

## Bottleneck matrices and combinatorial Perron parameters for trees

The notion of combinatorial Perron value  was introduced in [1]. We continue the study of this parameter and also introduce a new parameter $\pi_e(M)$ which gives a new lower bound on the spectral radius of the bottleneck matrix $M$ of a rooted tree. We prove a bound on the approximation error for $\pi_e(M)$. Several properties of these two parameters are shown. These ideas are motivated by the concept of algebraic connectivity. A certain extension property for the combinatorial Perron value is shown and it allows us to define  a new center concept for caterpillars. We also compare computationally this new center to the so-called characteristic set, i.e., the center obtained from algebraic connectivity.

Joint work with Lorenzo Ciardo and Geir Dahl.

### Reference

1. E. Andrade, G. Dahl, Combinatorial Perron Values of Trees and Bottleneck Matrices, Linear and Multilinear Algebra 65 (12) (2017): 1-19.
Natália Bebiano

## Maximum entropy principle for certain non-Hermitian Hamiltonians

The consideration of non-Hermitian Hamiltonians in quantum physics in the last decades gave rise to an intense research activity in physins and mathematics. In this talk, we focus in the extension for this setup of classical results on thermodynamical inequalities, and related topics.

Luca Gemignani
Department of Computer Science, University of Pisa

## Accelerating the Sinkhorn-Knopp iteration by Arnoldi-type methods

The task of balancing a nonnegative matrix by positive diagonal matrices reduces to find the fixed point of a nonlinear operator. The Sinkhorn-Knopp algorithm provides a simple iterative method for solving this problem but it can exhibit a very slow convergent behavior even in deceivingly simple cases. In this talk it is shown that the fixed point problem can also be recast as a constrained nonlinear multiparameter eigenvalue problem. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the diagonal transformations. Numerical results illustrate that our novel methods accelerate significantly the convergence of the customary Sinkhorn-Knopp iteration for matrix balancing especially in the case of clustered dominant eigenvalues. This is a joint work with Alessio Aristodemo from the University of Pisa.

Charles Johnson
College of William and Mary

## Tridiagonal Matrices

We give recent results on the spectral theory of (nonsymmetric) tridiagonal matrices over a field.

Eleftherios Kastis
Lancaster University

## The first order flexibility of a crystal framework

A crystal framework $C$ in $\mathbb R^d$ is a bar-joint framework $(G, p)$, where $G = (V, E)$ is a countable simple graph and $p : V \to \mathbb R^d$ is an injective translationally periodic placement of the vertices as joints $p(v)$. Given a crystal framework, the space of all complex infinitesimal flexes is the vector space of $C^d$-valued velocity fields on the joints of the framework that satisfy the first-order flex condition for every bar. This countable set of equations gives rise to an expanded version of the adjacancy matrix of a graph, called the rigidity matrix.

Due to the periodic structure of the framework the flex space is invariant under the natural translation operators and is closed with respect to the topology of coordinatewise convergence. In this talk, we shall indicate some new methods from analysis and commutative algebra. In particular, we generalise Lefranc’s spectral synthesis theorem for the case $r = 1$ to a vector-valued setting, to obtain that the flex space of a crystal framework is the closed linear span of flexes which are vector-valued polynomially weighted geometric multi-sequences.

This is joint work with Professor Stephen Power.

Vinod Kumar
Kurukshetra University

## Application of $f$-lacunary statistical convergence to approximation theorems

The concept of $f$-lacunary statistical convergence which is, in fact, a generalization of lacunary statistical convergence, has been introduced recently by Bhardwaj and Dhawan (Abstr. Appl. Anal., 2016). The main object of this paper is to prove Korovkin type approximation theorems using the notion of $f$-lacunary statistical convergence. A relationship between the newly established Korovkin type approximation theorems via $f$-lacunary statistical convergence, the classical Korovkin theorems and their lacunary statistical analogs has been studied. A new concept of $f$-lacunary statistical convergence of degree $\beta$ ($0 \lt \beta \lt 1$) has also been introduced and as an application a corresponding Korovkin type theorem is established.

Jean Jacques Loiseau
LS2N, CNRS - Université de Nantes

## Controlled invariance for systems over rings or semirings

The concept of controlled invariance is an important tool to formulate a number of control problems for linear systems, like model matching, trajectory tracking or disturbance decoupling. It leads to constructive methods and solutions for finite dimensional time-invariant systems, that are based on the computation of an invariant region. We are interested into the extension of these methods for systems over a ring or a semiring. Our motivation is to address the control of systems defined over a network, that are met in particular in communication processes and in production management. Such systems are in general characterized by time delays and constraints on the state and the control, that are handled using specific classes of systems, namely time delay systems or max-plus linear systems. These models can be seen as linear systems with coefficients on a ring or semiring of operators, and some control problems can be formulated in terms of cone or polyhedral invariance. The solution of these problems consists in the identification of invariant cones or polyhedra. Further, the controlled invariance of a region is equivalent to the existence of a feedback that makes it invariant for the closed-loop system, and the computation of such a feedback is the key step toward the implementation of the solution. In general, this feedback is not linear but solution of a linear system of equations. We identify some operator rings and semirings for which the existence of such a feedback is checkable, and illustrate the results on examples from production management.

Ivica Nakić
University of Zagreb

## Structured Lyapunov Functions and Dissipativity in LTI dynamical networks

In the talk I will address connections between structured storage or Lyapunov functions of a class of interconnected systems (dynamical networks) and dissipativity properties of the individual systems. I will present a result which states that if a dynamical network, composed as a set of linear time invariant (LTI) systems interconnected over an acyclic graph, admits an additive quadratic Lyapunov function, then the individual systems in the network are dissipative with respect to a (nonempty) set of interconnection neutral supply functions. Each supply function from this set is defined on a single interconnection link in the network. Specific characterizations of neutral supply functions will be presented which imply robustness of network stability/dissiptivity to removal of interconnection links.

Ana Pedro Lemos Paião

## Sufficient optimality conditions for delayed optimal control problems

We give answer to an open question by proving sufficient optimality conditions for optimal control problems with time delays in the state and control variables. In the proof of our main results, we transform delayed optimal control problems to equivalent non-delayed problems. This allows us to use well-known theorems that ensure sufficient optimality conditions for non-delayed optimal control problems. Finally examples are given with the purpose to illustrate the obtained results.

Katerina Sklyar
Institute of Mathematics, University of Szczecin

## Mappability of control systems to linear systems with analityc matrices

We consider a non-linear non-stationary control system  and study the problem of mappability of this system to a given linear system with analytic matrices by the change of variables. We give necessary and sufficient conditions of such mappability which in particular consist in coinsiding od some invariants of the systems. These invariants are  the certain meromorphic functions that characterize the system. We also study the question of description of such invariants in the case of two-dimentional systems.

Fernando de Terán

## Uniqueness of solution of generalized Sylvester-like equations with rectangular coefficients

We provide necessary and sufficient conditions for the generalized $\star$-Sylvester matrix equation, $AXB + CX^\star D = E$, to have exactly one solution for any right-hand side $E$. These conditions are given for arbitrary coefficient matrices $A, B, C, D$ (either square or rectangular) and generalize existing results for the same equation with square coefficients [1]. We also review the known results regarding the existence and uniqueness of solution for generalized Sylvester and $\star$-Sylvester equations. The contents of this talk have been recently published in [2].

References:

1. F. De Terán, B. Iannazzo,  Uniqueness of solution of a generalized $\star$-Sylvester matrix equation, Linear Algebra Appl., 493 (2016), pp. 323--335.
2. F. De Terán, B. Iannazzo,, F. Poloni, and L. Robol, Uniqueness of solution of generalized Sylvester-like equations with rectangular coefficients, Linear Algebra Appl., 542 (2018), pp. 501--521.
Izhar Uddin
Jamia Millia Islamia, Department of Mathematics

## Proximal Point Algorithm in Geodesic Spaces

In 1970, Martinet (Inform. Rech. Oper. 4 (1970), 154-158) introduced the well known Proximal Point Algorithm, popularly known as PPA which serves as an important tool to solve the minimization problem. Later on, in 1976, Rockafellar (SIAM J. Control Optim. 14 (1976), 877-898) studied PPA to prove the convergence of a solution of the convex minimization problem in the framework of Hilbert space. In 2013, Bacak (Israel. J. Math. 194 (2013), 689-701) introduced the proximal point algorithm in $\operatorname{CAT}(0)$ space. Note that the method has been modified so that it converges strongly by Cholamjiak (Optim. Lett. 9 (2015), 1401-1410) using the Halpern procedure. In this paper, we study proximal point algorithms for solving optimization problems in $\operatorname{CAT}(0)$ space via Thakur Iteration scheme.

David Wenzel
TU Chemnitz

## What is the shape of an operator acting on matrices?

As is known, the singular values of a linear operator give a quite handy description of it. The unit sphere of vectors is mapped to an elliptical set with half-axes of these lengths. However, this connection becomes kind of useless when the arguments are again matrices. An operator like $A:\mathbb{C}^n\to\mathbb{C}^n,x\mapsto Ax$ is represented well by the singular values as the components of the image vector $Ax$ are bounded. When considering $A:\mathbb{C}^{n\times n}\to\mathbb{C}^{n\times n},X\mapsto AX$, one may, of course, regard $X$ as an vector in $\mathbb{C}^{n^2}$, but this would neglect its true nature. Luckily, there are general estimates for $AX$ involving the singular values of $A$ and $X$.

The situation gets trickier when looking at bilinear maps like the commutator $(X,Y)\mapsto XY-YX$. The analog of the initially given image then could be:  What is the possible range of singular values if the ones of $X$ and $Y$ are given? Of course, it is possible to re-interpret such a map as a linear one that operates on tensor products. But, just looking at the Hilbert-Schmidt norms already unveils that these numbers are exaggerated by far due to the inclusion of lots of new arguments that are not coming from original input $X$ and $Y$.

So, we want to explore this question by investigating exemplary maps via snapshots. In the vector case the discreteness comes from the components of an orthogonal basis. This role shall be shifted to the rank. Consequently, we ask for the shape of the singular spectrum range when the input matrices are restricted to have singular value vectors $(a,0,\ldots)$, $(b,b,0,\ldots)$, and so on.

Michał Wojtylak
Jagiellonian University in Kraków

## Linear algebra properties of dissipative Hamiltonian descriptor systems

We will study the linear algebra properties of matrix pencils that are associated with linear time-invariant dissipative Hamiltonian descriptor systems of the form $E\dot x=\left(J-R\right)Qx.$ where $J,R\in\mathbb{C}^{n,n}$, $E,Q\in\mathbb{C}^{n,m}$, $m\leq n$,  $J^*=-J$ and $R^*=R\geq 0$. The following properties are shown: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues on the imaginary axis are semisimple, the index is at most two, and there are restrictions for the possible left and right minimal indices.

The talk is based on a joint paper with C. Mehl and V. Mehrmann with the same title (SIMAX 2018).

João Xavier
Instituto Sistemas e Robótica, LARSyS, Instituto Superior Técnico, Universidade de Lisboa

## Distributed Learning Algorithms for Big Data

Modern datasets are increasingly collected by teams of agents that are spatially distributed: sensor networks, networks of cameras, and teams of robots. To extract information in a scalable manner from those distributed datasets, we need distributed learning. In the vision of distributed learning, no central node exists; the spatially distributed agents are linked by a sparse communication network and exchange short messages between themselves to directly solve the learning problem. To work in the real-world, a distributed learning algorithm must cope with several challenges, e.g., correlated data, failures in the communication network, and minimal knowledge of the network topology. In this talk, we present some recent distributed learning algorithms that can cope with such challenges. Although our algorithms are simple extensions of known ones, these extensions require new mathematical proofs that elicit interesting applications of probability theory tools, namely, ergodic theory.

Anam Zahra
Pakistan Institute of Engineering and Technology

## Fractional Electrical Circuits under controllability and observability

In this talk, we will discuss few electrical circuits under fractional derivative state space formulation. after their descriptor fractional order mathematical model we will analyse their controllability and observability conditions with the help of Rank conditions which are obtained with the help of Gramian matrices and Drazin inverse.