Functional Analysis and Applications Seminar 

It will be presented a characterization of Fredholm and invertibility properties for Wiener-Hopf-Hankel operators with semi-almost periodic Fourier symbols and acting between $L^2$ Lebesgue spaces, based on the mean motions and geometric mean values of the almost periodic representatives of the Fourier symbols at minus and plus infinity. Additionally, a formula for the Fredholm index is derived, as well as some conditions for the invertibility of the operators.

We will discuss the current state of the factorization problem for almost periodic matrix functions and the consequences it has for the spectral theory of related Toeplitz operators. Some open problems will be presented as well.

Kokilashvili, Paatashvili, and Samko proved in 2005 that the Cauchy singular integral operator is bounded on variable Lebesgue spaces with Khvedelidze weights on arbitrary Carleson curves. We show that if the Carleson curve satisfies, in addition, a so-called logarithmic whirl condition at each point, then every semi-Fredholm operator in the Banach algebra of singular integral operators with matrix piecesise continuous coefficients is Fredholm.

This talk is based on a joint work with V. G. Kravchenko and J. S. Rodríguez. We consider a generalization of the results of a recent work by the authors concerning scalar singular integral operators with a backward Carleman shift, allowing more general coefficients, bounded measurable functions on the unit circle. The main purpose is to obtain, for singular integral operators with a backward shift and bounded measurable coefficients, an operator factorization from which the Fredholm characteristics, like the kernel and the cokernel, can be described. The main tool is the factorization of matrix functions. In the course of the analysis performed for that class of operators several useful representations are obtained which permit, in particular, to completely characterize the set of invertible operators in that class, providing explicit examples of such operators.

It is well known that when dealing with (pure) singular integral operators on the unit circle with coefficients belonging to a decomposing algebra of continuous functions, a factorization of the symbol induces a factorization of the original operator, which is a representation of the operator as a product of three singular integral operators where the outer operators in that representation are invertible. In our seminar we will show a similar operator factorization for the case of singular integral operators with a backward shift. We also show that the factorization of the considered operators is related to a (special) factorization in a algebra of block diagonal matrix functions and that such operator factorization is also possible for other classes of singular integral operators, namely those including either a conjugation operator or a composition of a conjugation with a forward shift operator.

Hölder spaces of variable order $\lambda(x)$ varying from point to point are well known. Meanwhile, it is also possible to consider generalized Hölder spaces with a characteristic $\omega(h)=\omega(x,h)$ which may also depend on the point $x$, similarly to the case of a Musielak-Orlicz space with the Young function $\Omega(x,u)$ depending on the point $x$. Since the Zygmund-Bary-Stechkin classes of characteristics $\omega$ are described in terms of the so called index numbers of $\omega$, in this generalization we arrive at indices depending on the parameter $x$ (this parameter in general may belong to an arbitrary abstract set, in applications this set may be a set in metric measure spaces). In this talk we consider properties of such parameter dependent index numbers, one of the main points being the study of conditions under which the Zygmund type inequality for $\omega(x,\cdot)$ is uniform with respect to the parameter. We shall discuss an application to measuring local dimensions of a metric measure space at a point $x$ and give an application to the study of generalized Hölder spaces on the unit sphere in the Euclidean space.

We start from a short survey of known results in Harmonic Analysis in Lebesgue spaces $L_p$ on metric measure spaces with doubling condition (homogeneous spaces) in the case of constant $p$, this case having a long history, and in the case of variable $p=p(x)$, for this case only a single result on non-weighted boundedness of the maximal operator being recently obtained. After that we give a new result on the weighted boundedness of maximal Hardy-Littlewood operator on homogeneous metric measure spaces, for variable $p(x)$. We prove this result under certain sufficient condition which we call an "ersatz" of the Muckenhoupt condition. This sufficient condition nevertheless coincides with the necessary Muckenhoupt condition when $p$ is constant. A special class of weights is also considered, which includes "radial type" weights oscillating between two power functions (Zygmund-Bary-Stechkin type functions). In this case a stronger statement on the weighted boundeness is obtained. In connection with the weighted boundedness, we also introduce a new notion of lower and upper local dimensions of metric measure spaces. The talk is based on the joint work with Prof. Vakhtang Kokilashvili.

We study asymptotics of block Toeplitz matrices generated by matrix symbols with entires in a generalized Hölder space or in the closure of smooth functions in the generalized Hölder space. We specify the speed of convergence in the Szegö-Widom limit theorems and refine corresponding results of Böttcher and Silbermann for standard Hölder spaces. Wiener-Hopf factorization in decomposing algebras plays a crucial role in our investigation.

We consider a nonstationary scattering of plane waves by a wedge. It is assumed that the incident wave does not depend on the coordinate parallel to the the edge of the wedge, so the problem is planar. Also we assume that, beginning with a certain time instant depending on a spatial position of the point, the incident wave is periodic in time with the frequency \(\omega\) in each point of the space. Let the profile of the wave be such that the incident wave has the front ahead of which it is zero. Therefore the incident wave establishes a harmonic vibration at any point of the complement of the wedge with the frequency \(\omega\). The main goal is to prove that the amplitude of the solution to the corresponding mixed problem for the D'Alembert equation with initial data determined by the incident wave, tends to the solutions of the classical stationary diffraction problem. Thus, these classical solutions can be represented as the limiting amplitudes of the solutions to the non-stationary problem, i.e. the Limiting Amplitude Principle holds. It is proved for the Dirichlet and Neumann boundary conditions and for Dirichlet-Neumann boundary conditions only for the right angle.

We propose writing partial differential equations on a hypersurface in cartesian coordinates of the ambient space instead of more customary local coordinates and the Riemannian metric tensor of the underlying surface. This seemingly trivial idea simplifies the form of many classical differential equations on the surface (Laplace-Beltrami, Lamé, Maxwell etc.), which turn out to have constant coefficients, and enables more transparent proofs of Korn's inequalities, tightly connected with solvability and uniqueness of some boundary value problems. The obtained results are applied to the Dirichlet and Neumann boundary value problems for the Laplace-Beltrami operator, for its square, and for the elasticity Lamé operators, describing thin shells in the form of an open smooth hypersurface with smooth boundary. An explicit Green formula is derived and it is proved that the Dirichlet boundary value problems has a unique solution in the Sobolev space of weak solutions while the Neumann boundary value problems are solvable under the usual orthogonality constraints on the data.

It is shown that the $2\times 2$ symbols in a class of exponentials of nilpotent matrices can be reduced, by splitting some rational factors, to a very simple normal form. A meromorphic factorization [1] of these symbols is thus naturally defined and by approaching the problem of transforming it into a generalized factorization from a point of view different from that of [1], we study the invertibility and the Fredholm properties of the Toeplitz operators with symbols in that class. These results simplify and generalize those obtained in [2]. This is a joint work with Cristina Câmara, IST, Lisboa.

1. Câmara, M. C., Lebre, A., Speck, F.-O. Generalised factorisation for a class of Jones form matrix functions. Proc. Roy. Soc. Ed., 123A (1993) 401-422.
2. Câmara, M. C., Malheiro, M. T. Wiener-Hopf factorization for a group of exponentials of nilpotent matrices. Linear Algebra Appl. 320(1-3) (2000) 79-96.

In this talk, I will give an introduction to index theory in the non-commutative geometry framework. $K$-theory for $C^{*}$-algebras will be developed, and it will be shown how one can associate $K$-theory classes to Fredholm operators, so that the Fredholm index arises as a map in $K$-theory. We will see also how this approach leads to generalized Fredholm indices, which do not take integer values, but instead take values in some $K$-theory group. One example of particular importance is that of families of operators.

We give a survey on generalized Krein algebras and Wiener-Hopf factorization in these algebras. We discuss the role of generalized Krein algebras in the asymptotic theory of Toeplitz determinants. We pay special attention to some cases of "insuffucient smoothness", when the Szegö-Widom limit theorem requires a higher order correction involving additional terms and regularized operator determinants. This talk is based on a joint work with Albrecht Böttcher and Bernd Silbermann.

The talk is devoted to studying pseudodifferential operators with non-regular symbols and their applications to singular integral operators with shifts. Algebras of singular integral operators with discrete subexponential groups of shifts are studied on weighted Lebesgue spaces provided that the contour, the weight, the coefficients and the shifts are slowly oscillating. Applying a local-trajectory method and a theory of Mellin pseudodifferential operators with non-regular symbols, we construct Fredholm symbol calculi for the mentioned algebras of singular integral operators with shifts and establish corresponding Fredholm criteria.

Firstly, we show mapping properties (within countably normed spaces) of several boundary integral operators acting on polygons (e.g. the weakly singular single layer potential operator for the Laplacian and the corresponding double layer potential operator). Our analysis of the boundary integral equations is based on Mellin techniques and uses the Mellin symbols of the integral operators. Functions (with corner singularities but) belonging to countably normed spaces can be approximated very efficiently by the $hp$-version of the boundary element method when refining the mesh size $h$ and increasing the polynomial degree $p$. Secondly, we analyze the Dirichlet problem for the Laplacian in a polygonal domain. Applying Mellin techniques to the boundary integral equation we show that the solution has a decomposition into regular and singular parts which blow up at certain exceptional angles. We derive a modified decomposition which depends continuously on the angle and can be used efficiently for boundary element computations.

The goal of the talk is to present the boundedness criteria for singular integrals in weighted Lebesgue spaces with variable exponent. The following topics will be discussed: boundedness of the generalized singular integrals on Carleson curves arising in the theory of I. N. Vekua's generalized analytic functions; sufficient conditions and examples of a couple of weights ensuring two-weight estimates for singular integrals and maximal functions in variable Lebesgue spaces; applications to the Dirichlet boundary value problem in "bad" domains and to the problem of the mean summability of Fourier trigonometric series in non-standard, two-weight setting.

We study Toeplitz operators in a weighted Bergman space on the unit disc with a power type weight related to the boundary of the disc. We deal with special symbols connected to the three types of hyperbolic geometry in the unit disc (elliptic, parabolic and hyperbolic pencils). That is, in each of the mentioned three cases the symbols are constant on geodesics orthogonal to the trajectories forming a pencil. The spectrum of each of the Toeplitz operator seems to be quite accidental, the definite tendency starts appearing only as the exponent of the weight tends to infinity. The correspondence principle (F. Berezin) suggests that the limit set of those spectra has to be strictly connected with the range of the initial symbol. This is definitely true for continuous symbols. Given a continuous symbol a, the limit set of spectra does coincide with the range of a. The new effects appear when we consider more complicated symbols. In particular, in the case of piecewise continuous symbols the limit set coincides with the range of a together with the line segments connecting the one-sided limit points of piecewise continuous symbol. Note that these additional line segments may essentially enlarge the limit set comparing to the range of a symbol.

In this talk the problem of existence and calculation of solutions to Lax equations that define finite-dimensional integrable systems is studied. The method presented is based on Wiener-Hopf factorization and related Riemann-Hilbert problems on Riemann surfaces. The idea behind the method was first proposed by Semenov-Tian-Shansky but has never been applied in a situation where a nontrivial function factorization is required. An example of a dynamical system associated with an elliptic curve is given.

The linear-fractional problem is a generalization of the linear Riemann problem that includes the (non-linear) factorization problem. In case of normal type it can be equivalently reduced to a family of homogeneous linear vector Riemann problems by space foliation and adequate substitutions. Moreover these problems are equivalent to systems of non-homogeneous Toeplitz equations with special data. The reduced problem can be solved by matrix factorization in various cases. This research is based upon joint work with S. V. Rogosin.

We consider a class of Riemann-Hilbert problems with triangular symbols. Our investigation is devoted to the existence and calculation of a solution, in the form of an almost periodic polynomial. The Fourier spectrum of a solution of this kind is a subset of a particular additive group. A necessary and sufficient condition for the existence of a solution is obtained. Indeed, it is a simple condition on the Fourier spectrum of the matrix symbol. Explicit solutions are also determined, for different classes of Riemann-Hilbert problems, which are determined once again by the matrix symbol.