Functional Analysis and Applications Seminar 

A description of the Calkin algebra generated by multiplication, Wiener-Hopf and Hankel operators has been known for some time. More recently, those results were generalized to the Calkin algebra generated by a flip, multiplication and convolution operators, with the use of non-commutative Gelfand theories, homogenization and flip-elimination techniques. Unfortunately the flip-elimination technique fails when studying, to get stability conditions, the local algebras obtained when a finite section or spline projection sequence is added, in \(L_p\). In this talk the problem will be discussed, and possible solutions given.

In this talk we will speak about orthogonal polynomials, focusing our presentation on the so called inverse problems. Several connections with different branches of mathematics will be pointed out. In particular, we will consider some problems in the framework of approximation theory involving Sobolev orthogonal polynomials as well as applications to the spectral theory of Jacobi operators.

Our aim in this talk is to deal with the boundedness of the Hardy-Littlewood maximal operator on Musielak-Orlicz-Morrey spaces. As an application of the boundedness of the maximal operator, we establish Sobolev's inequality for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces.

A C*-algebra \(A\) is semiprojective if, roughly speaking, any approximate homomorphism from \(A\) to some other C*-algebra is close to an actual homomorphism. (There are several different ways to make this precise, leading to several different concepts.) There are not very many semiprojective C*-algebras, but the ones that do exist (and the fact that they are semiprojective) play an important role in the theory. Examples of semiprojective C*-algebras include finite dimensional C*-algebras, the algebra of continuous functions from the interval or circle to a finite dimensional C*-algebra, the Cuntz algebras and some of their generalizations, and the full C*-algebras of free groups. Now suppose that a compact group \(G\) acts on \(A\). We say that \(A\) is equivariantly semiprojective if, roughly speaking, whenever \(B\) is another C*-algebra with an action of \(G\), then every approximately equivariant approximate homomorphism from \(A\) to \(B\) is close to an exactly equivariant homomorphism. We prove that finite dimensional C*-algebras are equivariantly semiprojective, as well as Cuntz algebras with certain special actions. One of the many applications of semiprojectivity is to classification theorems. In the classification of purely infinite simple C*-algebras, semiprojectivity is used to replace asymptotic morphisms with homomorphisms. We expect equivariant semiprojectivity to play the same role in the classification of actions of compact groups on purely infinite simple C*-algebras.

Suppose we are given an orientation preserving diffeomorphism (shift) of the semi-axis onto itself with the only fixed points zero and the point at infinity. We establish Fredholmn criteria for a class of singular integral operator with shift under the assumptions that the coefficients and the derivative of the shift are bounded and continuous on the open semi-axis and may admit discontinuities of slowly oscillating type at zero and at infinity. In this talk we briefly discuss the proof of the sufficiency part (presented at WOTCA 2010) and give a more detailed account on the proof of the necessity part. This talk is based on the joint work with Yuri Karlovich and Amarino Lebre.

A Fredholm criterion and an index formula are established for the singular integral operator with a shift associated to the Haseman boundary value problem with oscillating data on star-like curves in the weighted Lebesgue space setting. The study is based on the theory of Mellin pseudodifferential operators. Applications to index calculations for singular integral operators with a shift will be discussed.

In this talk, we consider Hankel operators, associated with a complex valued function $f$ which is square integrable with respect to a given measure, in a Hilbert space of meromorphic functions related to a moment sequence $s$ . We investigate the boundedness and compactness of such operators in connection with the growth and the regularity of the sequence $s$.

Wavelets and Bergman spaces appear to be completely disjoint objects, but they are not. In this talk we will explain why. Indeed, wavelet theory offers a real variable approach to Bergman spaces that goes beyond the classical toolkit of analytic functions. In particular, this approach yields new results concerning interpolation and sampling in spaces of polyanalytic functions, using ideas from application-oriented mathematics.

The problem of providing a complete description of the facial structure of the unit ball in a symmetric complex Banach space has been studied for thirty years and has often seemed intractable. This talk will begin with a history of the problem and end with its solution. This represents joint work with Francisco Fernandez-Polo, Christopher Hoskin, and Antonio Peralta.

Current conjectures and almost The theory of Random Matrices has seen a large development during the last decade with connections to many different areas in Mathematics. In my talk I will elaborate, by example, only one connection with Operator Theory. Simply put, a random matrix is a matrix whose entries are chosen at random. It is described by the class of matrices considered and by the underlying probability distribution. The main interest is in the eigenvalues of the random matrices and their asymptotics when the matrix size is large. We consider certain ensembles of non-hermitian complex random matrices and the linear statistics of their eigenvalues. The linear statistics is a random variable which is a sum over some test function on the eigenvalues. In the large $n$ limit, the linear statistics is expected to obey some kind of central limit theorem. The key to dealing with the asymptotics of the linear statistics is that, under certain conditions, their probability distribution can be expressed in terms of the determinants of matrices are the Hadarmad of a Toeplitz and a Hankel matrix. In the cases under consideration, the Toeplitz structure is the dominating one, and a Limit Theorem will be established, which resembles the classical Szego Limit Theorem for Toeplitz determinants. However, depending on the underlying Random Matrix Ensemble, two cases with a different kind of asymptotics are identified.

We consider spatial discretizations by the finite section method of the restricted group C*-algebra of a finitely generated discrete group, which is represented as a concrete operator algebra via its left-regular representation. Special emphasis is paid to the quasicommutator ideal of the algebra generated by the finite sections sequences and to the stability of sequences in that algebra. For both problems, the sequence of the discrete boundaries plays an essential role. Finally, for commutative groups and for free non-commutative groups, the algebras of the finite sections sequences are shown to be fractal.

Let $U$ be a complex domain Möbius-equivalent to a disk and let $j$ be a non zero integer. The talk will focus on explicit representation of the poly-Bergman projection of order $j$, in terms of the canonical two-dimensional singular integral operators. One also shows how the Lebesgue space $L^2(U;\mathrm{dA})$ decomposes on the true poly-Bergman spaces, where $\mathrm{dA}$ is the element of Lebesgue area measure. The poly-Bergman kernels of $U$ are explicitly calculated.

We will recall the notion of the numerical range (also known as the field of values), outline its known beautiful properties, and discuss some generalizations and results obtained recently. In particular, the ratio field of values will be treated in detail.

The talk is devoted to different methods of studying the Fredholmness in algebras of singular integral operators with discrete groups of shifts. The local-trajectory method, applications of spectral measures and irreducible representations, the limit operators techniques and the pseudodifferential operators approach will be demonstrated for algebras of one- and two-dimensional singular integral operators with shifts.

The effect on the propagation of acoustic waves, from a slow motion in the medium where the propagation occurs, is usually very small. An important exception is, however, when the mean flow of the motion is unstable. There could then be an interaction between the acoustic wave and the mean flow and this interaction could be strong. One such instability occurs when the mean flow separates from a waveguide boundary at a sudden area expansion or the exit. The talk describes the formulation and solution of mathematical models describing the acoustic-flow interaction at such discontinuities. This formulation is of wave scattering nature and is solved with Wiener-Hopf techniques, which are particularly suited for the inclusion of unstable wave parts. Of particular importance is to classify the types of instabilities: a) convective instability, when the stationary solution exists and b) absolute instabilities when a stationary solution does not exist. For a particular model of the flow, the vortex sheet model, the stationary solution exists but is an ultra distribution. As an introduction to the talk, basic stability theory is recalled. At the end, predictions from simulations with the theory are presented together with confirming experimental results.

The aim of this seminar is to present our progress in the explicit generalized factorization of some special matrix-valued functions. We construct an algorithm to obtain the solutions of certain singular integral equations related with a self-adjoint operator. Using these solutions we construct another algorithm to determine an effective factorization of the matrix functions. Using Mathematica 6.0 symbolic computation package we implement the two algorithms on a digital computer, automating the factorization process as a whole. We present some examples, both in the real line and in the unit circle, obtained with the Mathematica application. The presentation is based on a joint work with Viktor G. Kravchenko.

We study the weighted \(p\to q\)-boundedness of multi-dimensional Hardy type operators in Morrey spaces for a class of almost monotonic weights. The obtained results are applied to a similar weighted \(p\to q\)-boundedness of the Riesz potential operator. The conditions on weights, both for the Hardy and potential operators are necessary and sufficient in the case of power weights. In the case of more general weights we provide separately necessary and sufficient conditions.

Continuous, bounded representations of a group on a Banach space give rise, via integration, to convolution operators on that space. It turns out that for several natural representations and a large class of groups, any such convolution operator will have empty residual spectrum. In this talk, I shall present some of the background to these questions, and then discuss some recent work on this theme, focusing mainly on the case of discrete groups. The root of all these results is an old observation of Kaplansky, concerning left-invertible elements of the group von Neumann algebra.

We study potential operators in Hölder-type spaces over uniform domains in the Euclidean space and show that they map the subspace of functions vanishing at the boundary into the improved Hölder-type space The problem in the study is related to the absence of the so called cancellation property for domains, our proofs being based on a special treatment of the potential of a constant function and the usage of the uniformity of domains (Jones domains). The talk is based on a joint work with Lars Diening.

In this seminar, I first would like to introduce some fundamental theory for linear transforms in the framework of Hilbert spaces and its general applications with very concrete typical examples. Secondly, I will propose a new method for inversions of bounded linear operators on Hilbert spaces with discretization. The contents will be given by the following key words: Reproducing kernel, Pythagorean theorem, inversion of the linear systems, inverse problem, inverse heat conduction, real inversion of the Laplace transform, analytic extension, observation data, non-linear system, non-linear transform, generalized inverse, Tikhonov regularization, singular integral equations.