Abstracts

A separated graph is a pair $(E,C)$, where $E$ is a directed graph, $C=\bigsqcup _{v\in E^ 0} C_v$, and $C_v$ is a partition of $r^{-1}(v)$ (into pairwise disjoint nonempty subsets) for every vertex $v$. In recent years, separated graphs have been used to provide combinatorial models of several structures, often related to dynamical systems. This can be understood as a generalization of the common use of usual directed graphs in symbolic dynamics. I will survey some of these developments, including the failure of Tarski’s dichotomy in the setting of topological actions, the construction of a family of ample groupoids with prescribed type semigroup, and the modeling of actions on the Cantor set.

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.

More than two decades after the publication of Joseph L. Taylor’s article Functions of several noncommuting variables (1973), the analytic machinery he introduced in this paper found, thanks to Dan V. Voiculescu’s work from the late ’90s and early 2000s, extremely fruitful (and sometimes unexpected) applications to free probability and random matrix theory. These applications led to numerous important results, for instance isomorphism results for von Neumann algebras, regularity results for tuples of non-commuting random variables, or asymptotic behavior results for random matrices. In this talk we will present the intimate connection between Taylor’s noncommutative functions and Voiculescu’s noncommutative distributions, via their noncommutative analytic transforms. We will show that the free independence of two tuples of noncommutative random variables is equivalent to a simple functional equation satisfied by their generalized Cauchy-Stieltjes transforms. Various versions of this functional equation have been used as well in random matrix theory for computing joint distributions of random matrices: we will present a few sample results. Finally, we will show some regularity results for joint distributions of free variables, together with the main ideas of their proofs.

In random matrix theory, some of the fundamental eigenvalue distributions can be expressed as tau functions defined by Fredholm determinants, so \[\tau (x) =\det (I+K_x)\] where $K_x$ is an integral operator of trace class on $L^2(0, \infty )$. In important cases, $K_x$ is given by a Hankel integral operator $\Gamma_\Phi$ such that $\Phi$ is a matrix-valued symbol function. Megretskii, Peller and Treil characterized the bounded and self-adjoint Hankel operators $\Gamma$ up to unitary equivalence, and also characterised those $\Gamma$ such that $\Phi$ may be realised from a linear system $(-A,B,C)$ with state space $L^2(0, \infty )$, so $\Phi (x) =Ce^{-xA}B$.

In this talk, we discuss how to realise Laguerre and Whittaker functions from explicit linear systems, and show how the corresponding tau functions satisfy the nonlinear ordinary differential equation Painlevé VI. We also mention periodic linear systems, which can be used to realise Jacobi’s elliptic theta function $\vartheta_1$.

From the linear system $(-A,B,C)$ one can introduce an algebra of operators on the state space, subject to groups of deformations. From this, one can derive systems of nonlinear partial differential equations satisfied by tau, such as the Kadomstev-Petviashvili equation.

References

• G. Blower and Y. Chen, Kernels and point processes associated with Whittaker functions, Journal of Mathematical Physics 57 (2016), 093595, 17 pages.
• G. Blower, On the tau function for orthogonal polynomials and matrix models, Journal of Physics A: Mathematical and Theoretical 44 (2011), 285202, 31 pages.
• G. Blower and S. L. Newsham, On tau functions associated with linear systems, 2012 arXiv: 1207.2143.

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.

Recently, S. Chandler-Wilde and D. Hewett have proposed a boundary integral equation approach for studying scattering problems involving fractal structures, in particular planar screens which are fractal or have a fractal boundary. This led them to consider, e.g., subspaces of Bessel-potential spaces like $H^s_F:=\{u\in H^s(\mathbb R^n): \operatorname{supp}\,u \subset F\}$ when $F$ is a closed subset of $\mathbb R^n$ and $\widetilde{H}^s(\Omega):=\overline{{\cal D}(\Omega)}^{H^s(\mathbb R^n)}$ when $\Omega$ is an open subset of $\mathbb R^n$ and, together with A. Moiola, study some properties of such spaces.

As examples of questions of interest in this regard we have the following:

• For which $s\in\mathbb R$ and $\Omega$ open do we have $\widetilde{H}^s(\Omega)=H^s_{\overline{\Omega}}$?
• For which $s\in\mathbb R$ and $K$ compact with empty interior but with positive Lebesgue measure do we have $H^s_K \not= \{ 0 \}$? Or $H^s_K = \{ 0 \}$?
• When is $H^t_F$ dense in $H^s_F$ for $F$ closed and $t\gt s$?

In this talk I shall report on this and also on some answers to which we have arrived, using some current function spaces techniques, during our recent collaboration project. Besides, since the techniques involved in general work in a more general framework than the one presented above, I take the opportunity to dwell also on some relevant aspects of the modern theory of function spaces of Besov and Triebel-Lizorkin type which might also be useful in other settings.

We derive from variational principles a class of stochastic partial differential equations and show the existence of their solutions.

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.

We consider two examples of boundary value problems (BVPs) on hypersurfaces: heat conduction by an "isotropic" media, governed by the Laplace equation and bending of elastic "isotropic" media governed by Láme equations. The boundary conditions are classical Dirichlet-Neumann mixed type. The domain $\Omega^{\varepsilon }:=\mathcal{C}\times (-\varepsilon ,\varepsilon )$ is of thickness $2\varepsilon$. Here $\mathcal{C}\subset \mathcal{S}$ is a smooth subsurface of a closed hypersurface $\mathcal{S}$ with smooth nonempty boundary $\partial \mathcal{C}$.

The object of the investigation is what happens with the above mentioned mixed boundary value problem when the thickness of the layer converges to zero. It is shown that the corresponding BVPs converge in the sense of $\Gamma$-convergence to a certain BVPs on the mid surface $\mathcal{C}$: The BVP for the Laplace equation converges to the BVP for the Dirichlet BVP for the Laplace-Beltrami equation, while for the Láme equation we get a new form of BVP for the shell equation.

The suggested approach is based on the fact that the Laplace and Láme operators are represented in terms of Günter’s tangential and normal (to the surface) derivatives. For example, the laplace operator $\Delta_{\Omega^{\varepsilon }}=\partial _{1}^{2}+\partial _{2}^{2}+\partial _{3}^{2}$ is represented as the sum of the Laplace-Beltrami operator on the mid-surface and the square of the transversal derivative: $\Delta _{\Omega ^{\varepsilon }}T = \sum\limits_{j=1}^{4}\mathcal{D}_{j}^{2}T=\Delta _{\mathcal{C}}T+\partial _{t}^{2}T$.

The work is carried out in collaboration with T. Buchukuri and G. Tephnadze (Tbilisi).

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.

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.

We will discuss some examples of zeta-regularised spectral determinants of elliptic operators, focusing on the effect of the spatial dimension and the order of the operator. The former case will be illustrated by the harmonic-oscillator, while for the latter we consider polyharmonic operators on bounded intervals. In both cases we obtain the first terms in the asymptotic expansion as the dimension/order approaches infinity.

Bishop operators $T_{\alpha}$ acting on $L^2[0,1)$ were proposed by E. Bishop in the fifties as possible operators which might entail counterexamples for the Invariant Subspace Problem. In this talk we will show some of their main properties and prove, by means of arithmetical techniques along with a theorem of Atzmon, that the set of irrationals $\alpha\in (0,1)$ for which $T_\alpha$ is known to possess non-trivial closed invariant subspaces is enlarged, extending previous results by Davie, MacDonald and Flattot.

(Joint work with F. Chamizo, M. Monsalve-López and A. Ubis)

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

1. 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.
2. M. A. Kaashoek and F. van Schagen, The inverse problem for Ellis-Gohberg orthogonal matrix functions, Integr. Equ. Oper.Theory 80 (2014), 527-555.
3. 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.

Let $\mathcal{B}_{p,w}$ be the Banach algebra of all bounded linear operators on the weighted Lebesgue space $L^p(\mathbb{T},w)$ with $p\in(1,\infty)$ and a Muckenhoupt weight $w\in A_p(\mathbb{T})$ which is locally equivalent at open neighborhoods $u_t$ of points $t$ in the unit circle $\mathbb{T}$ to weights $W_t$ for which the functions $\tau\mapsto(\tau-t)(\ln W_t)'(\tau)$ are quasicontinuous on $u_t$, and let $PQC$ be the $C^*$-algebra of all piecewise quasicontinuous functions on $\mathbb{T}$. The Banach algebra \[ \mathfrak{A}_{p,w}={\rm alg}\big\{aI,S_\mathbb{T}:\ a\in PQC\big\}\subset\mathcal{B}_{p,w} \] generated by all multiplication operators $aI$ by functions $a\in PQC$ and by the Cauchy singular integral operator $S_\mathbb{T}$ is studied. A Fredholm symbol calculus for the algebra $\mathfrak{A}_{p,w}$ is constructed and a Fredholm criterion for the operators $A\in\mathfrak{A}_{p,w}$ in terms of their Fredholm symbols is established by applying the Allan-Douglas local principle, the two idempotents theorem and a localization of Muckenhoupt weights $W_t$ to power weights by using quasicontinuous functions and Mellin pseudodifferential operators with non-regular symbols.

Further, for $w=1$, the Fredholmness is studied for the Banach algebra \[ \mathfrak{B}_p={\rm alg}\big\{aI,S_\mathbb{T},U_g:\ a\in PQC,\ g\in G\big\}\subset\mathcal{B}_p\] being the extension of the Banach algebra $\mathfrak{A}_p$ by the isometric shift operators $U_g:f\mapsto |g'|^{1/p}(f\circ g)$ for $g\in G$, where $G$ is a subexponential (or amenable for $p=2$) group of orientation-preserving homeomorphisms $g$ of $\mathbb{T}$ onto itself, with piecewise slowly oscillating derivatives $g'$, which acts topologically freely on $\mathbb{T}\setminus\Lambda^\circ$, and $\Lambda^\circ$ is the interior of a nonempty closed set $\Lambda\subset\mathbb{T}$ composed by all common fixed points for all shifts $g\in G$. The study is based on two different local-trajectory methods (for the Banach and Hilbert space settings), with involving spectral measures, a lifting theorem and Mellin pseudodifferential operators with non-regular symbols. The results obtained for the algebra $\mathfrak{B}_p$ essentially depend on the structure of the set of fixed points for shifts $g\in G$ on $\mathbb{T}$.

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.

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

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

The Hilbert transform is a singular integral operator that gives access to harmonic conjugate functions via a convolution of boundary values. This operator is trivially a bounded linear operator in the space of square integrable functions. This is no longer obvious if we introduce a positive weight with respect to which we integrate the square. In this case, conditions on the weight need to be imposed, the so-called characteristic of the weight, both necessary and sufficient for boundedness. It is a delicate question to find the exact way in which the operator norm grows with this characteristic. Interest was spiked by a classical question surrounding quasiconformality and the Beltrame equation.

We give a brief historic perspective of the developments in this area of "weights" that spans about twenty years and that has changed our understanding of these important classical operators.

We highlight a probabilistic and geometric perspective on these new ideas, giving dimensionless estimates of Riesz transforms on Riemannian manifolds with bounded geometry.

Let ${\mathsf A}$ be a family of operators on a Hilbert space $H$ (for a concrete example, the set of Toeplitz operators $T(a)$ with $a \in C({\mathbb T})$ on $H = l^2$) and let $(P_n)$ be a sequence of projections on $H$ of finite rank with $P_n \to I$ strongly (e.g., $P_n : l^2 \to l^2, (x_k)_{k \in {\mathbb N}} \mapsto (x_1, \ldots, x_n, 0, 0, \ldots)$). We consider $(P_n A P_n)$ as an approximation sequence for $A \in {\mathsf A}$. A typical question in numerical analysis is whether this sequence is stable. To answer this question it is helpful to study the $C^*$-algebra generated by all sequences $(P_n A P_n)$ with $A \in {\mathsf A}$ (e.g., the algebra ${\mathcal S}({\mathsf T}(C))$ generated by all sequences $(P_n T(a) P_n)$ with $a \in C({\mathbb T})$). Several concepts were developed to study algebras of approximation sequences arising in this way. Two of these (compactness and fractality) will occur in this talk.

Compact sequences play a role comparable to compact operators. For example, if $K$ is compact on $l^2$, then the sequence $(P_n K P_n)$ belongs to the algebra ${\mathcal S}({\mathsf T}(C))$, and it is considered as a compact element of that algebra (there are other compact sequences in ${\mathcal S}({\mathsf T}(C))$ as well). Fractality is a property of algebras of approximation sequences $(A_n)$ which implies good convergence properties of spectral quantities related with the $A_n$ (e.g., the convergence of the norms $\|A_n\|$ and of the pseudospectra $\sigma_\varepsilon (A_n)$). Both concepts are related by the fact that the ideal of the compact sequences in a fractal algebra has a surprisingly simple structure: it is a dual algebra, i.e. a sum of algebras isomorphic to $K(H)$.

In this talk, I consider algebras which are not fractal, but close to fractal algebras in the sense that every restriction has a fractal restriction. Typical examples of such algebras are the algebra of the finite section discretization of block Toeplitz operators and algebras resulting from the discretization of the algebras $C(X, \, {\mathsf A})$, the ${\mathsf A}$-valued continuous functions on a Hausdorff compact $X$. We will discuss conditions which guarantee that these algebras again have a nice structure: they are isomorphic to a continuous field, and their compact sequences form an algebra with continuous trace.

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.

Dilation theory is a paradigm for understanding a general class of objects in terms of a better understood class of objects, by way of exhibiting every general object as “a part of” a special, well understood object.

In the first part of this talk I will discuss both classical and contemporary results and applications of dilation theory in operator theory. Then I will describe a dilation theoretic problem that we got interested in very recently: what is the optimal constant $c = c_{\theta,\theta'}$, such that every pair of unitaries $U,V$ satisfying $VU = e^{i\theta} UV$ can be dilated to a pair of $cU', cV'$, where $U',V'$ are unitaries that satisfy the commutation relation $V'U' = e^{i\theta'} U'V'$? I will present the solution of this problem, as well as a new application (which came to us as a pleasant surprise) of dilation theory to the continuity of the spectrum of the almost Mathieu operator from mathematical physics. 

Based on a joint work with Malte Gerhold. 
 

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.

Started in 1983-05 this research enjoyed a revival in 2015 by a paper titled Wiener-Hopf factorization through an intermediate space, in which an alternative to the cross factorization theorem was proposed that fits better with various applications. In general operator factorizations generate a certain “middle space” in a natural way that is related with important properties of the corresponding general or concrete Wiener-Hopf operator. We report about this paper [1], expose concerning applications [2], and some consequences [3,4].

1. Speck, Frank-Olme; Wiener-Hopf factorization through an intermediate space. Integral Equations Oper. Theory 82, No. 3, 395-415 (2015).
2. Speck, Frank-Olme; A class of interface problems for the Helmholtz equation in $\mathbb{R}^n$. Math. Meth. Appl. Sciences 40, No. 2 (2017), 391-403 .
3. Boettcher, Albrecht; Speck, Frank-Olme; On the symmetrization of general Wiener-Hopf operators. J. Operator Theory 76, No. 2, 335-349 (2016).
4. Speck, Frank-Olme; On the reduction of general Wiener-Hopf operators. To appear.

The numerical range $W(A)$ (a.k.a. the field of values, or the Hausdorff set) of a linear operator $A$ acting on a Hilbert space $\frak H$ is defined as the range of the mapping \[ f_A \colon x\mapsto \left\langle Ax,x \right\rangle \] on the unit sphere of $\frak H$. Its history goes back to celebrated papers by Toeplitz, [5] and Hausdorff, [1], in which the convexity of $W(A)$ was established.

We will give a brief overview of some other properties and applications of the numerical range, obtained since then. The following two topics will be discussed in more detail:

• Continuity properties of the (multivalued) inverse of $f_A$; in particular, strong continuity of $f_A^{-1}$ on the interior of $W(A)$.
• Normalized numerical range \[ W_N(A)=\{\left\langle Ax,x \right\rangle \left\|Ax\right\|\cdot\left\|x\right\|\colon x\in{\mathfrak H}, Ax\neq 0\}, \] and its relation with the Davis-Wielandt shell of $A$.

These parts of the talk are based mostly on [2] and [3, 4], respectively.

References

1. F. Hausdorff, Der Wertvorrat einer Bilinearform, Math. Z. 3 (1919), 314-316.
2. B. Lins and I. M. Spitkovsky, Inverse continuity of the numerical range map for Hilbert space operators, arXiv:1810.04199v1 (2018), 1-12.
3. B. Lins, I. M. Spitkovsky, and S. Zhong, The normalized numerical rangeand the Davis-Wielandt shell, Linear Algebra Appl. 546 (2018), 187-209.
4. I. M. Spitkovsky and A.-F. Stoica, On the normalized numerical range, Operators and Matrices 11 (2017), no. 1, 219-240.
5. O. Toeplitz, Das algebraische Analogon zu einem Satze von Fejér, Math. Z. 2 (1918), no. 1-2, 187-197.

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

By the classical Brown, Halmos (1964) result, there is no commutative $C^\ast$-algebra generated by Toeplitz operators, with non-trivial symbols, acting on the Hardy space $H^2(S^1)$, while there are only two, rather trivial, commutative Banach algebras generated by Toeplitz operators. For one of them symbols are analytic, and are conjugate analytic, for the other.

At the same time, as it was observed recently, there are many non-trivial commutative $C^\ast$-algebras generated by Toeplitz operators, acting on the Bergman space over the unit disk. Moreover, for a multidimensional case of the weighted Bergman space $\mathcal{A}^2_{\lambda}(\mathbb{B}^n)$, apart of a wide variety of commutative $C^\ast$-algebras, there exist many commutative Banach algebras, all of them are generated by Toeplitz operators with symbols from different specific classes.

The aim of the talk is to clarify the situation for a multidimensional Hardy space $H^2(S^{2n-1})$ case.

We present an universal approach that permits us to unhide and describe both commutative $C^\ast$ and Banach algebras generated by Toeplitz operators on $H^2(S^{2n-1})$, as well as to describe some non-commutative $C^\ast$-algebras. In the latter case we characterize, among others, their irreducible representations and spectral properties of the corresponding Toeplitz operators.

We explore some closed range type properties of Bergman Toeplitz operators with unbounded symbols, and use this to show few interesting connections to problems in complex and functional analysis, and problems related to other operators. The results we will present include, for example, a characterization of Fredholm Toeplitz operators with $\operatorname{BMO}^1$ symbols and a characterization of invertible Toeplitz operators with unbounded nonnegative symbols, in case their Berezin transforms are bounded and of vanishing oscillation.