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

- Pere Ara

Universitat Autònoma de Barcelona, Spain ## Separated graphs and dynamics

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.

- Serban Belinschi

CNRS - Institut de Mathématiques de Toulouse, France ## Analytic transforms of noncommutative distributions

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.- Gordon Blower

Lancaster University, UK ## Linear systems in random matrix theory

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.

- G. Blower and Y. Chen, Kernels and point processes associated with Whittaker functions,
- António Caetano

Universidade de Aveiro, Portugal ## Function spaces techniques in problems of scattering by fractal screens

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.

- Ana Bela Cruzeiro

Instituto Superior Técnico, Portugal ## On some stochastic partial differential equations obtained by a variational approach

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

- Roland Duduchava

University of Georgia, Tbilisi, Georgia ## Boundary value problems on hypersurfaces and $\Gamma $-convergence

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).

- Pedro Freitas

Instituto Superior Técnico, Portugal ## Spectral determinants of elliptic operators: dependence on spatial dimension and order of the operator

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.

- Eva Gallardo

Universidad Complutense de Madrid, Spain ## Invariant subspaces for Bishop operators and beyond

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)

- Yuri Karlovich

Universidad Autónoma del Estado de Morelos, México ## Algebras of singular integral operators with piecewise quasicontinuous coefficients and non-smooth shifts.

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 $\mathbb{T}$.

- Stefanie Petermichl

Université de Toulouse, France ## Change of measure

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.

- Steffen Roch

Technische Universität Darmstadt ## On quasifractal algebras

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.

- Orr Moshe Shalit

Technion, Israel ## Dilation theory: fresh directions with new applications

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.

- Frank-Olme Speck

Instituto Superior Técnico, Portugal ## Advances in general Wiener-Hopf factorization

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].- Speck, Frank-Olme;
*Wiener-Hopf factorization through an intermediate space*. Integral Equations Oper. Theory 82, No. 3, 395-415 (2015). - 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 . - Boettcher, Albrecht; Speck, Frank-Olme;
*On the symmetrization of general Wiener-Hopf operators*. J. Operator Theory 76, No. 2, 335-349 (2016). - Speck, Frank-Olme;
*On the reduction of general Wiener-Hopf operators*. To appear.

- Speck, Frank-Olme;
- Ilya Spitkovsky

New York University in Abu Dhabi, UAE ## One hundred years of... numerical range

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

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

- Nikolai Vasilevski

CINVESTAV, Mexico ## Algebras generated by Toeplitz operators on the Hardy space $H^2(S^{2n-1})$

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.

- Nina Zorboska

University of Manitoba, Canada ## Toeplitz operators on the Bergman space with $\operatorname{BMO}^p$ symbols and the Berezin transform

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.