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

- Joseph Ball

Virginia Tech University, USA ## Input/state/output linear systems and their transfer functions: from single-variable to multivariable to free noncommutative function theory

Given a system matrix $\left[ \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \right] \colon

\left[ \begin{smallmatrix} {\mathcal X} \\ {\mathcal U} \end{smallmatrix} \right] \to

\left[ \begin{smallmatrix} {\mathcal X} \\ {\mathcal Y} \end{smallmatrix} \right]$ (where ${\mathcal X}$ is the*state space*, ${\mathcal U}$ is the*input space*, ${\mathcal Y}$ is the*output space*), we associate the input/state/output linear system \begin{equation} \tag{1}\Sigma \colon \begin{cases} x(n+1) & = A x(n) + B u(n), \quad x(0) = x_0, \\ y(n) & = C x(n) + D u(n). \end{cases} \end{equation} One can solve the recursion and apply the $Z$-transform \[\{ w(n) \}_{n \in {\mathbb Z}_+} \mapsto \widehat w(\lambda) : = \sum_{n=0}^\infty w(n) \lambda^n \] to convert the system equations to functional form \[ \begin{cases}\widehat x(z) = (I - \lambda A)^{-1} x_0 + \lambda (\lambda I - a)^{-1} B \widehat u(\lambda), \\ \widehat y(z) = {\mathcal O}_{C,A}(\lambda) x_0 + T_\Sigma(\lambda) \widehat u(\lambda) \end{cases}\] where ${\mathcal O}_{C,A}(\lambda): = C (I - \lambda A)^{-1}$ is the*observability operator*and $T_\Sigma(\lambda) = D + C + \lambda C (I - \lambda A)^{-1} B$ is the*transfer function*for the system $\Sigma$. Central issues of interest for engineers are:**Realization:**Which functions $F(\lambda)$ can be realized as $F(\lambda) = T_\Sigma(\lambda)$ for some $\Sigma$? What is the uniqueness for such a realization?**Internal stability:**When is it the case that $x(n) \to 0$ as $n \to \infty$ for any $x_0$ when $\{u(n)\}_{n \in {\mathbb Z}_+}$ is set equal to $0$?**Performance:**When is it the case that $T_\Sigma$ has $H^\infty$-norm $\sup\{ \| T_\Sigma(\lambda) \| \colon \lambda \in {\mathbb D} \}$ at most $1$?

For all these questions, it is also important to have computationally effective algorithms or solution criteria (e.g., Linear-Matrix-Inequality conditions expressed directly in terms of the system-matrix entries $A,B,C,D$ for the last two questions). We review the results for the classical 1-D systems as in (1) and indicate how the formalism extends to (i) the setting of multidimensional systems with transfer functions now holomorphic functions of several complex variables on a domain $\Omega \subset {\mathbb C}^d$, and (ii) the setting of structured noncommutative linear systems with transfer functions equal to formal power series or equivalently free noncommutative functions (in the sense of Kaliuzhnyi-Verbovetskyi and Vinnikov) on a noncommutativedomain contained in $\cup_{n=1}^\infty ({\mathbb C}^{n \times n})^{d} $. An interesting feature is that the results for the setting (i) are only partial generalizations while the results for setting (ii) are compelling, complete extensions of the classical case.

- Albrecht Böttcher

Technische Universität Chemnitz, Germany ## Lattices from equiangular tight frames

As I have the honor to give an ILAS lecture, I take the liberty to leave the field of genuine operator theory and to move into linear algebra. The talk is about the question when certain matrices do generate a lattice, that is, a discrete subgroup of some finite-dimensional Euclidean space, and if this happens, which good properties these lattice have. The matrices considered come from equiangular tight frames. I promise a nice tour through some basics of equiangular lines, tight frames, and lattice theory. We will encounter lots of interesting vectors and matrices and enjoy some true treats in the intersection of discrete mathematics and finite-dimensional operator theory.

- Kenneth Davidson

University of Waterloo, Canada ## Noncommutative Choquet theory

We introduce a new framework for noncommutative convexity. We develop a noncommutative Choquet theory and prove an analogue of the Choquet-Bishop-de Leeuw theorem. This is joint work with Matthew Kennedy.

- Ruy Exel

Universidade Federal de Santa Catarina, Brasil ## Statistical Mechanics on Markov spaces with infinitely many states

We shall begin by reviewing a compactification of the Markov space for an infinite transition matrix, introduced by Marcelo Laca and the speaker roughly 20 years ago. Given a continuous potential we will then consider the problem of characterizing the conformal measures on that space. Along the way a somewhat unexpected but very natural generalization of Renault’s notion of approximately proper equivalence relations will force its way into the picture leading up to the construction of a natural étale groupoid whose quasi-invariant measures we shall also discuss observing that they are examples of what may be seen as generalized DLR (Dobrushin-Lanford-Ruelle) measures. In the context of the Markov shifts mentioned above we will then explore the connections between conformal and DLR measures.

- Hans Feichtinger

University of Vienna, Austria ## Classical Fourier Analysis and the Banach Gelfand Triple

It is the purpose of this presentation to explain certain aspects of Classical Fourier Analysis from the point of view of

*distribution theory*. The setting of the so-called*Banach Gelfand Triple*$(S_0,L^2,S_0')(\mathbb{R}^d)$ starts from a particular Segal algebra $S_0(\mathbb{R}^d)$ of continuous and Riemann integrable functions. It is Fourier invariant and thus an extended Fourier transform can be defined for $S_0'(\mathbb{R}^d)$, the space of so-called*mild distributions*. Any of the $L^p$-spaces with $1 \leq p \leq \infty$ contains $S_0(\mathbb{R}^d)$ and is embedded into $S_0'(\mathbb{R}^d)$.We will show how this setting of

*Banach Gelfand triples*(resp.*rigged Hilbert spaces*) allows to provide a conceptual appealing approach to most classical parts of Fourier analysis. In contrast to the Schwartz theory of tempered distributions it is expected that the mathematical tools can be also explained in more detail to engineers and physicists.- Rien Kaashoek

Vrije Universiteit, Amsterdam, Netherlands ## Inverting structured operators and solving related inverse problems

In this talk we will analyse two classical theorems from a higher point of view. The first theorem is the famous Gohberg-Heinig inverse theorem for self-adjoint finite Toeplitz operator matrices. The general setting that will be presented involves a Hilbert space operator $T$ and a contraction $A$ such that the compression of $T-A^\ast TA$ to the null space of the defect operator $(I-AA^\ast)^{1/2}$ is the zero operator. For such an operator $T$ the problem is: when is $T$ invertible and when $T$ is invertible do we have a formula for its inverse? The answers are remarkable similar to those of the classical problem (see [1]).

The second classical theorem we shall be dealing with is the famous Szegö-Kreĭn inverse theorem for orthogonal matrix polynomials. In our general setting the data of the inverse problem are Hilbert space operators

\begin{align*} A :\mathcal{X}\to& \mathcal{X}, \ \|A\| \leq 1, \quad B:\mathcal{Y}\to \mathcal{X},\quad C : \mathcal{X}\to \mathcal{Y},\\ &\overline{\operatorname{Im} C}=\mathcal{Y} \text{ and } I-AA^*=C^*C. \end{align*} Given these data the problem is: under what conditions on $B$ does there exists a self-adjoint operator $T$ on $\mathcal{X}$ such that $TB=C^\ast $ and the compression of $T-A^\ast TA$ to the null space of $C$ is zero.The inverse problem for Ellis-Gohberg orthogonal Wiener class functions on the unite circle fits into this setting. We shall present the solution of the latter problem for matrix-valued Wiener class functions, and, if time permits, we shall also discuss the twofold version of the inverse problem. For several examples the problem is still open.

### References

- A. E. Frazho and M. A. Kaashoek, A contractive operator view on an inversion formula of Gohberg-Heinig, in:
*Topics in Operator Theory I. Operators, matrices and analytic functions*, OT**202**, Birkhäuser Verlag, Basel, 2010, pp. 223-252. - M. A. Kaashoek and F. van Schagen, The inverse problem for Ellis-Gohberg orthogonal matrix functions,
*Integr. Equ. Oper.Theory***80**(2014), 527-555. - S. ter Horst, M. A. Kaashoek, and F. van Schagen, The discrete twofold Ellis-Gohberg inverse problem,
*J. Math. Anal. Appl.***452**(2017), 846-870.

- A. E. Frazho and M. A. Kaashoek, A contractive operator view on an inversion formula of Gohberg-Heinig, in:
- Igor Klep

Univerza v Ljubljani, Slovenia ## Bianalytic maps between matrix convex sets

Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry, semidefinite programming, control theory and signal processing. LMIs with (dimension free) matrix unknowns, called free LMIs, are central to the theories of completely positive maps and operator algebras, operator systems and spaces, and serve as the paradigm for matrix convex sets. The feasibility set of a free LMI is called a free spectrahedron.

In this talk, the bianalytic maps between a very general class of

*ball-like*free spectrahedra (examples of which include row or column contractions, and tuples of contractions) and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form. They are all highly structured rational maps we call convexotonic. In particular, this yields a classification of automorphism groups of ball-like free spectrahedra. The results depend on new tools in free analysis to obtain fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of free spectrahedra.- Lars-Erik Persson

UiT - The Arctic University of Norway ## My life with Hardy and his inequalities

First of all I will shortly describe some facts concerning the fascinating prehistory and history of Hardy-type inequalities.

After that I will present some fairly new discoveries how some Hardy-type inequalities are closely related to the concept of convexity. I will continue by presenting some facts from the further development of Hardy-type inequalities as presented in remarkable many papers and also some monographs (see e.g. [1] and cf. also [2]). Moreover. I will present some very new results and raise a number of open questions.

### References

- A. Kufner, L. E. Persson and N. Samko, Weighted Inequalities of Hardy type, World Scientific, Second edition, New Jersey-London-etc., 2017
- L. E. Persson, Lecture Notes, College de France, Pierre-Louis Lions’ Seminar, November 2015. 1

- Peter Semrl

Univerza v Ljubljani, Slovenia ## Automorphisms of effect algebras

Let $H$ be a Hilbert space. By $E(H)$ we denote the effect algebra on $H$, that is, the set of positive operators on $H$ that are bounded by the identity. Effect algebras are important in mathematical foundations of quantum mechanics. There are quite a few operations and relations defined on $E(H)$ which play a significant role in different aspects of quantum theory. Besides the usual partial ordering $\le$, the most important are the sequential product defined by $A \circ B = A^{1 /2} B A^{1/2}$, the orthocomplementation given by $A^\perp = I - A$, and the coexistency. Two effects $A,B \in E(H)$ are said to be coexistent if there exist effects $E,F,G$ such that $A= E+G$, $B= F+G$, and $E+F+G \in E(H)$.

Mathematical physicists are interested in symmetries, that is, bijective maps on quantum structures that preserve certain relations and/or operations in both directions. We will present some recent results on symmetries of effect algebras.

### References

- G. P. Gehér and P. Semrl, Coexistence preservers on Hilbert space effect algebras, preprint.
- G. Ludwig,
*Foundations of quantum mechanics, Vol. I*, Springer-Verlag, 1983. - L. Molnár, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces.
*Lect. Notes Math*.**1895**, Springer-Verlag, 2007. - L. Plevnik and P. Semrl, Automorphisms of effect algebras, Oper. Theory Adv. Appl.
**271**, Birkhauser/Springer, Cham, 2018, 361-387. - P. Semrl, Comparability preserving maps on Hilbert space effect algebras,
*Comm. Math. Phys.***313**(2012), 375-384. - P. Semrl, Symmetries of Hilbert space effect algebras,
*J. London Math. Soc.***88**(2013), 417-436. - P. Semrl, Automorphisms of Hilbert space effect algebras,
*J. Phys. A***48**(2015), 195301,18pp.

- Bernd Silbermann

Technische Universität Chemnitz, Germany ## Invertibility Issues for Toeplitz plus Hankel operators

Toeplitz and Hankel operators appear in various ﬁelds of mathematics, physics and statistical mechanics and they have been rigorously studied. The theory of Toeplitz plus Hankel operators $T(a) + H(b)$ is less developed. Nevertheless, Fredholm theory for operators with piecewise continuous generating functions acting on Hardy spaces $H^p(\mathbb{T})$ or on $l^p$-spaces has been developed. On the other hand, their kernels, cokernels and invertibility remain little studied because the operators $T(a) + H(b)$ own some features known for block Toeplitz operators, that is the latter have only in rare cases eﬃcient invertibility conditions and kernel descriptions. In the last years a few concepts have been developed to study these problems. It is the aim of this talk to give some overview in case that the generating functions $a$ and $b$ fulﬁll the so-called matching condition, that is \[a(t)\widetilde{a}(t) = b(t)\widetilde{b}(t) ,\quad t \in \mathbb{T},\] where $\widetilde{c}(t) = c(\frac{1}{t})$. This condition leads to a transparent theory which covers many interesting operators. Moreover, close relatives of $T(a) + H(b)$, such as Wiener-Hopf plus Hankel operators, can also be treated by this theory.

- Christiane Tretter

Universität Bern, Switzerland ## Spectra and essential spectra of nonselfadjoint operators

In this talk techniques to obtain reliable information on the spectrum and essential spectra of nonselfadjoint operators will be presented.