Functional Analysis and Applications Seminar 

The sharp upper bound for the Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability $p\geq 1$) on fractal $d$-sets is obtained: $\min \{ d+1-s,d/s \} $, where $s\in (0,1]$ denotes the smoothness parameter. In particular, when passing from $d\geq s$ to $d \lt s$ there is a change of behaviour from $ d+1-s $ to $d/s$ which implies that even highly nonsmooth functions defined on cubes in $\mathbb{R}^n$ have not so rough graphs when restricted to, say, rarefied fractals.

Classically, for a given equation $Ax=b$ and a sequence of compact projections $𝒫=(P_n)$ which converges strongly to the identity one studies the sequence of (truncated) equations $P_{n}A{P}_n{x}_n=P_{n}b$ in order to find approximate solutions for the initial problem. The theory behind that idea is heavily based on the interactions between compactness, Fredholmness and strong convergence. In the first part of this talk we now turn the table in a sense, and we take a sequence $𝒫=(P_n)$ (called approximate projection) as a starting point for the definition of appropriate substitutes which we call $𝒫$-compactness, $𝒫$-Fredholmness and $𝒫$-strong convergence. On the one hand, this adapted framework permits to develop a theory that mimics the classical one and that provides very similar results on the applicability of the projection method, the stability, and on the asymptotics of norms, condition numbers or pseudospectra. On the other hand, it can be applied to much more general settings since it is detached from the fixed classical notions. The second part picks up the approximation of pseudospectra in more detail. We particular demonstrate how Hansens concept of $(N,\epsilon )$-pseudospectra can be generalized to the Banach space case and how this may help to deal with the phenomenon of jumping pseudospectra.

In this talk we introduce Grand Grand Morrey spaces, in the spirit of the so-called Grand Lebesgue spaces. We show the validity of a kind of reduction lemma, which is aplicable to a variety of operators, to reduce their boundedness in Grand Grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund operators in the framework of Grand Grand Morrey spaces.

Variable Toeplitz matrices, a generalization of the finite sections of a Toeplitz operator, have received increasing interest in recent years. They also appear under different names: generalized locally Toeplitz matrices, generalized convolutions or twisted Toeplitz matrices, with applications to discretized PDE, probability theory and statiscal mechanics. We will focus on sequences of variable- coefficient Toeplitz matrices with symbol on a class of the Wiener type. Using C*- algebras techniques, we will discuss the asymptotic behaviour of singular values, condition numbers and pseudospectrum, as well as the analogon of Fredolhm theory for sequences.

The talk is devoted to the Fredholm study of \(C^*\)-algebras of Bergman type operators with piecewise continuous coefficients and commutative groups of conformal mappings of a bounded simply connected plane domain onto itself.

In boundary problems for elliptic systems, namely through the method of layer potentials, one is often led to study invertibility of integral operators on the boundary. If the domain is sufficiently regular, classic Fredholm theory applies. On singular domains, however, the relevant operators are no longer compact. The main aim of this talk is to give a suitable replacement of classic Fredholm theory in the setting of domains with conical singularities. The key idea is to use the theory of pseudodifferential operators on Lie groupoids. In that respect, to a conical domain $\Omega$ we first associate a boundary groupoid $\mathcal{G}$ over a desingularization of $\partial \Omega$ and define the so-called layer potentials $C^\ast$-algebra, which turns out to be a good replacement for the ideal of compact operators. We use a representation of $\Psi(\mathcal{G})$ as bounded operators on suitable Sobolev spaces with weight at $\partial \Omega$ to give Fredholm criteria, reducing to ellipticity and invertibility of indicial operators on cones at each singularity.

The talk is based on joint work with Victor Nistor and Yu Qiao.

The modelling of diffraction of time-harmonic electromagnetic or acoustic waves from obstacles and screens leads to boundary value problems for the three-dimensional Helmholtz equation with Dirichlet, Neumann or other conditions on the boundary. A prominent example is the problem of diffraction from a quarter-plane in $\mathbb{R}^3$, which admits an explicit solution. In this paper the Dirichlet and Neumann problems for the three-quarter-plane are solved by an algebraic trick: the matricial coupling of operators associated to "dual" boundary value problems, a kind of abstract Babinet principle.

We first present some facts from the Spherical Harmonic Analysis, related to decompositions of functions into series of spherical harmonics and spherical convolution operators invariant with respect to rotations. Then we use some properties of spherical convolution operators to solve an integral equation over semishere in the n-dimensional Euclidean space which arises in a certain problem of aerodynamics. In this problem there is considered a rarefied medium of non-interacting point masses moving at unit velocity in all directions. Given the density of the velocity distribution, one easily calculates the pressure created by the medium in any direction. We consider the inverse problem: given the pressure distribution, determine the density. This leads to the problem of solving the above mentioned integral equation. In the "application part" the talk is based on a joint paper with Alexander Plakhov (Universidade de Aveiro).

We show that a pseudodifferential operator with symbol in the Hörmander class $S_{r,d}^{n(r-1)}$ is bounded on a reflexive variable Lebesgue space for a wide range of parameters r and d whenever the Hardy-Littlewood maximal operator is bounded. Further we prove a sufficient condition for the Fredholmness of a pseudodifferential operator with a symbol that slowly oscillates in the first variable and belongs to $S_{1,0}^0$. Both theorems generalize pioneering results by Rabinovich and Samko (IEOT, 2008) obtained for globally log-Hölder continuous variable exponents. This is a joint work with Ilya Spitkovsky.

Hecke algebras can be seen as an analogue of group algebras of quotient groups $G/H$ when $H$ is no longer a normal subgroup. They admit several canonical $C^{*}$-completions and when some of these coincide, and a maximal $C^{*}$-norm exists, there is a nice correspondence between $C^{*}$-representations of a Hecke algebra and unitary representations of $G$ generated by the $H$-fixed vectors. In this seminar I will give an overview of the theory of Hecke algebras and their various $C^{*}$-completions. I will also discuss some recent work on the existence of a maximal $C^{*}$-completion and the isomorphism problem among the remaining canonical completions, for several classes of Hecke algebras.

The algebraic structure of the complete lattice of weak*-closed inner ideals in a JBW*-triple is fundamental in describing both their mathematical properties and their physical applications. The talk will give an introduction to the subject and describe some recent developments.

A conjecture of Halmos proved by Choi and Li states that the closure of the numerical range of a contraction on a Hilbert space is the intersection of the closure of the numerical ranges of all its unitary dilations. We show that for $C_0(N)$ contractions one can restrict the intersection to a smaller family of dilations. This generalizes a finite dimensional result of Gau and Wu.

We give an introductory talk on a recent progress in the so-called "PT-symmetric quantum theory", in which the usual self-adjointness of observables requirement is replaced by their simultaneous Parity-Time invariance. The latter "often" implies that the spectrum is real and that the time evolution is unitary when reconsidered in a Hilbert space with appropriately changed inner product. The relevance of PT-symmetry has been suggested in various domains of physics, however, so far, there has been no experimental evidence proving that quantum systems defined by PT-symmetric Hamiltonians do exist in nature. In this talk, inter alia, we propose a simple PT-symmetric interpretation of a class of Sturm-Liouville operators with non-Hermitian Robin-type boundary conditions as a (physical) perfect-transmission scattering problem. Moreover, we establish closed integral-type formulae for similarity transformations relating the non-Hermitian operators with self-adjoint Hamiltonians, succeed in writing down the latter as a simple integro-differential operator and also find the associated "charge conjugation" operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

Let ${B}_{p,w}$ denote the Banach algebra of all bounded linear  operators acting on the weighted Lebesgue space $L^p(\mathbb{R},w)$ where  $1 \lt p \lt \infty$ and $w$ is in a subclass of Muckenhoupt weights. We study the  Banach subalgebra ${A}_{p,w}$ of ${B}_{p,w}$ generated  by all convolution type operators of the form $a\mathcal{F}^{-1}b\mathcal{F}$  where $\mathcal{F}$ is the Fourier transform, the functions $a, b\in L^\infty (\mathbb{R})$ admit piecewise slowly oscillating discontinuities on  $\mathbb{R}\cup\{\infty\}$ and $b$ is a Fourier multiplier on $L^p(\mathbb{R},w)$. Applying results on commutators of pseudodifferential operators with non-regular symbols, the Allan-Douglas local principle and  the limit operators techniques, we construct a Fredholm symbol calculus and  obtain a Fredholm criterion for the operators $A\in {A}_{p,w}$ in  terms of their Fredholm symbols.

The talk is based on a joint work with I. Loreto Hernández.

A Fredholm symbol for a nonlocal operator C*-algebra \(B\) of singular integral operators, with an amenable discrete group of homeomorphisms having the same nonempty set of periodic points, is obtained. Using the spectral measure associated with a general isometric representation of \(B\) we will see how we can reduce the study of the Fredholmness in \(B\) to the study of the Fredholmness in a C*-algebra \(B_F\) of singular integral operators with a group of homeomorphisms with fixed points.

This talk is based on a joint work with M. A. Bastos and Y. Karlovich.

The existence of singularities of solutions of Lax equations is related to the kernel of a Toeplitz operator.The singularities are obtained from the study of an associated Riemann-Hilbert problem. An example is presented.

Let $a$ be a semi-almost periodic matrix function with the almost periodic representatives $b$ and $c$ at $-\mathrm{\infty }$ and $+\mathrm{\infty }$, respectively. Suppose $p(\cdot )$ is a slowly oscillating exponent such that the Cauchy singular integral operator $S$ is bounded on the variable Lebesgue space $L_p(\cdot )$. We prove that if the operator $aP+Q$ with $P=(I+S)/2$ and $Q=(I-S)/2$ is Fredholm on the variable Lebesgue space $L_p(\cdot )$, then the operators $bP+Q$ and $cP+Q$ are invertible on standard Lebesgue spaces $L_q$ and $L_r$ with some exponents $q$ and $r$ lying in the segments between the lower and the upper limits of $p(\cdot )$ at $-\mathrm{\infty }$ and $+\mathrm{\infty }$, respectively. This is a joint work with Ilya Spitkovsky.

Completeness of dilation systems $(f(n x))$ with $n \gt 0$ on the standard Lebesgue space $L_2(0,1)$ is considered for $2$-periodic functions $f$. We show that the problem is equivalent to an open question on cyclic vectors of the Hardy space $H_2(D_2^\infty)$ on the Hilbert multidisc $D_2^\infty$. Several simple sufficient conditions are exhibited, which contain however practically all previously known results (Wintner; Kozlov; Neuwirth, Ginsberg, and Newman; Hedenmalm, Lindquist, and Seip). The Riemann Hypothesis on zeros of the Euler zeta-function is known to be equivalent to a completeness of a similar but non-periodic dilation system (due to Nyman).

The so-called abc theorem for polynomials, also known as Mason's or Mason-Stothers' theorem, deals with nontrivial polynomial solutions to the Diophantine equation \(a+b=c\). It provides a lower bound on the number of distinct zeros of the polynomial abc in terms of the degrees of \(a\), \(b\) and \(c\). We prove some "local" \(abc\) type theorems for general analytic functions living on a (reasonably nice) bounded domain rather than on the whole plane. The estimates obtained are sharp, for any domain, and they imply a generalization of the original "global" \(abc\) theorem by a limiting argument.

The talk is devoted to asymptotic properties of certain matrix sequences called variable-coefficient Toeplitz matrix sequences (there are also different names for them like twisted Toeoplitz matrices or locally Toeplitz matrices). The main aim is to present Szegő-like theorems and to discuss further asymptotics.