Functional Analysis and Applications Seminar 

The notion of a reflexive linear space of operators is closely related with the invariant subspace problem for complex Banach spaces. There are several generalizations of this notion. One of them is \(k\)-reflexivity, where \(k\) is an arbitrary positive integer. One can show that a linear space of operators is \(k\)-reflexive if and only if it is an intersection of kernels of a set of elementary operators of length at most \(k\). Thus, it is natural to ask when is the kernel of a given elementary operator \(k\)-reflexive. We will present some results related to this question.

We will propose an integral equation analysis of electromagnetic scattering by waveguides based on the characterization of the Fredholm property and Douglis-Nirenberg ellipticity of corresponding integral operators. As a prototype, the scattering of time-harmonic electromagnetic waves by an infinitely long cylindrical orthotropic waveguide iris will be considered in detail. This will be done by using an orthotropic Maxwell system in a cylindrical waveguide iris for plane waves, imbedded in an isotropic infinite medium. This complicated 3D problem is successfully reduced to the solution of a simpler 2D problem. The unique solvability of the associated boundary value problems is proved and regularity of solutions is also obtained in Bessel potential spaces. Part of the talk is based on a joint work with R. Duduchava and D. Kapanadze.

The invertibility and Fredholmness of Toeplitz operators with $2\times 2$ matrix symbol $G$ can be studied in connection with some properties of a solution to a Riemann-Hilbert problem with coefficient $G$. We show that from this solution we can obtain formulas for the factors of a Wiener-Hopf factorization of $G$, when it exists, which allows us to construct a generalized inverse for the Toeplitz operator. This is the first of a pair of talks on this subject based on joint work with M. C. Câmara.

Let $X$ be a compact metric space with finite covering dimension, and equipped with a free minimal action of $Z^d.$ When $d=1,$ a fair amount is understood about the transformation group C*-algebra $C^{*}(Z^d,X),$ but for $d>1$ very little is known about it except when $X$ is the Cantor set. We describe how to prove, under an additional technical condition, that $C^{*}(Z^d,X)$ has strict comparison for positive elements. This condition says, roughly speaking, that the order on the Cuntz semigroup of $C^{*}(Z^d,X)$ is determined by the tracial states on $C^{*}(Z^d,X),$ which in this case all come from invariant probability measures on~$X.$ We use this result to deduce the more familiar condition, that the order on projections over $C^{*}(Z^d,X)$ is determined by the tracial states. However, we do not know how to prove this result without using the Cuntz semigroup. The technical condition is satisfied whenever $X$ is a smooth manifold and the action is via diffeomorphisms. In this talk, I will focus on the part of the proof involving the Cuntz semigroup, and how the Cuntz semigroup of a suitable ''large'' subalgebra of $C^{*}(Z^d,X)$ can be used to obtain information about the Cuntz semigroup of $C^{*}(Z^d,X)$ itself.

Siegfried Proessdorf studied with me, in 1987-89, the factorization of Daniele-Khrapkov matrices in spaces of Hoelder continuous functions. The work was motivated by a special matrix function that was found in diffraction theory by E. Meister in 1977 and "ad hoc factored" by A.R. Rawlins in 1981. It was recognized soon as an important example of a matrix function that admits an explicit generalized factorization (although not rationally reducible to a triangular matrix), and gave a great impact on factorization theory, developed in our research center at Lisbon. Recently diffraction by non-rectangular (but rational) wedges has shown an interesting connection with the existence of certain extension operators, either from half-lines into cones bordered by this half-line on one side, or into cones which contain this half-line in its interior up to the common vertex, such that the extended function is a weak solution of the Helmholtz equation. The existence of the latter extension operator is somehow equivalent to the inversion of a certain pure Hankel operator. In this lecture we focus on the relations between the above-mentioned operators and formulate some resulting open problems.

The dimensions of the kernels of two Toeplitz operators whose symbols satisfy certain relations are compared using simple linear algebraic arguments. The results are applied to obtain some Fredholm type properties for \(2\times 2\) symbols whose determinant admits a bounded factorization.

We prove the applicability of the finite section method to an arbitrary operator in the Banach algebra generated by the operators of multiplication by piecewise continuous functions and the convolution operators with symbols in the algebra generated by piecewise continuous and slowly oscillating Fourier multipliers on $L_p(\mathbb{R})$. This is a joint work with Alexei Karlovich and Pedro A. Santos. Preprint is available at http://arxiv.org/abs/0909.3821

We are concerned with the applicability of the finite sections method to operators belonging to the closed subalgebra of the algebra of the linear bounded operators acting on $L_p$, generated by operators of multiplication by piecewise continuous functions in $\mathbb{R}$ and operators of convolution by piecewise continuous Fourier multipliers. The usual technique is to introduce a larger algebra of sequences, which contains the special sequences we are interested and the usual operator algebra generated by the operators of multiplication and convolution. There is a direct relationship between the applicability of the finite section method for a given operator and invertibility of the corresponding sequence in this algebra. But, contrarily to the $C^*$ case and Banach analogue for Toeplitz operators, in our case several inverse-closedness problems must be solved.

We extend previous results for the Neumann boundary value problem in a convex cone $\Omega$ to the case where the boundary data are from the Sobolev trace space of order $-1/2 + \varepsilon , 0 \lt \varepsilon \lt 1/2$. We prove that for these boundary conditions the solution of the Helmholtz equation exists in the Sobolev space $H^s(\Omega)$ of order $s = 1 + \varepsilon$ ­ , is unique and depends continuously on the boundary data. Moreover we give the Sommerfeld representation for these solutions. This can be used to formulate explicit compatibility conditions on the data for regularity properties of the corresponding solution. We extend previous results for the Neumann boundary value problem in a convex cone $\Omega$ to the case where the boundary data are from the Sobolev trace space of order $-1/2 + \varepsilon , 0 \lt \varepsilon \lt 1/2$­ . We prove that for these boundary conditions the solution of the Helmholtz equation exists in the Sobolev space $H^s(\Omega)$ of order $s = 1 + \varepsilon$ ­ , is unique and depends continuously on the boundary data. Moreover we give the Sommerfeld representation for these solutions. This can be used to formulate explicit compatibility conditions on the data for regularity properties of the corresponding solution.

We will present some results and open problems concerning norms of some operators acting on a Hilbert space and even operator norms of matrices. The origin of this norm-computation problems is nonassociative algebra, which will also be briefly explained. We will consider some Jordan structures possesing models that can be represented as subspaces of the algebra of bounded operators on a Hilbert space. Even in this special cases norms of several algebraic operators (Jacobson-McCrimmon operator) are surprisingly difficult to compute, even in just two-dimensional case. This gives rise to several interesting problems, which can be understood without knowledge of nonassociative algebra and perhaps tackled by every student of functional analysis.

We consider generalized Morrey spaces with variable exponent $p(x)$ and a general function $\omega (x,r)$ defining the Morrey-type norm. In case of bounded domains we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev-Adams type for the Riesz-type potential operator, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities which do not assume any kind of monotonicity condition $\omega (x,r)$ in the variable. The latter makes the results new even in the case of constant $p$.

The talk is based on a joint paper with V. Guliev and M. Hajibayev.

We study the scattering of time-harmonic electromagnetic waves by a closed or an open surfaces surrounded by an anisotropic medium. The corresponding system is non-elliptic and the "electric" and "magnetic" boundary conditions are non-normal. For the complex valued (non-real) frequency parameter the Dirichlet type "electric" boundary value problem (BVP) is equivalently reduced to the Neumann type "magnetic" BVP, which in final analysis is reduced to an equivalent elliptic BVP in a space of functions orthogonal to a certain vector. Using potential method and tools of pseudodifferential operators the unique solvability and regularity results of the equivalent BVPs are proved for real valued, constant, positive definite and symmetric permeability and permittivity matrices. The talk is based upon joint work with O. Chkadua and D. Kapanadze.

In the first part of my talk I present a new method in order to obtain mapping properties of the boundary operators for slit diffraction occurring in Sommerfeld’s diffraction theory, covering two different cases of the polarisation of the light. This theory is entirely developed in the context of the boundary operators with a Hankel kernel and not based on the corresponding mixed boundary value problem for the Helmholtz equation. For a logarithmic approximation of the Hankel kernel we obtain explicit solutions together with certain regularity results. In the second part of my talk I use the Fourier method in order to present the boundary equations for the more general diffraction of light through an aperture in a plane screen. In this way we also obtain the solution of the diffraction problem behind the screen in terms of the Fourier transforms of the electrical boundary fields. However, these electrical boundary fields have to be determined from the singular boundary equations mentioned above. This is a common work with Dr. Norbert Gorenflo of the Beuth Hochschule für Technik Berlin, Germany.

We consider mixed boundary value problems for the scalar Helmholtz equation describing the diffraction by a (acoustically) soft strip and the diffraction by a (acoustically) soft circular disk, respectively. The talk focuses on representations of the solutions of the underlying boundary integral equations. Firstly, some of the speaker's earlier results are presented which are based on the reformulation as a Wiener-Hopf problem which differs from the hitherto known Wiener-Hopf formulations in that it is posed in the original space and not in the frequency space. Then, newer results are given, which admit series expansions of the solutions of the boundary integral equations in terms of functions and expansion coefficients that both are given in a closed form.

We investigate three-dimensional oscillation interface crack problems (ICP) for metallic-piezoelectric composite bodies with regard to thermal effects. We give a mathematical formulation of the physical problem when the metallic and piezoelectric bodies are bounded along some proper parts of their boundaries where interface cracks occur. By the potential method the ICP is reduced to an equivalent strongly elliptic system of pseudodifferential equations on manifolds with boundary. We study the solvability of this system in different function spaces and prove uniqueness and existence theorems for the original ICP. We analyse the regularity properties of the corresponding thermo-mechanical and electric fields near the crack edges and near the curves where different type boundary conditions collide. In particular, we characterize the stress singularity exponents and show that they can be explicitly calculated with the help of the principal homogeneous symbol matrices of the corresponding pseudodifferential operators. We expose some numerical calculations which demonstrate that the stress singularity exponents essentially depend on the material parameters.

The description of the system of $n$-subspaces of a linear space $V$ has become a classical problem of linear algebra. In particular, many works are devoted to the study of indecomposable fours of subspaces up to equivalence, representations of posets and other. We consider irreducible systems of n-subspaces of a Hilbert space $H$ up to unitary equivalence. With every subspace $H_i$ one can connect an orthoprojector $P_i$ on it and study irreducible $n$-tuples of orthoprojectors up to unitary equivalence. There is a well known list of all irreducible pairs of orthoprojectors (Halmos, 1965). But the description of irreducible three-tuples of orthoprojectors becomes a *-wild problem. So we add some conditions on subspaces of $H$ to describe them.

We consider an algebra of operator sequences containing, among others, the approximation sequences to convolution type operators on cones acting on the Lebesgue space. To each operator sequence $A_n$ we associate a family of operators $W_x(A_n)$ parametrized by $x$ in some index set. When all $W_x(A_n)$ are Fredholm, the so-called approximation numbers of $A_n$ have the $\alpha$ splitting property with $\alpha$ being the sum of the kernel dimensions of $W_x(A_n)$. Moreover, the sum of the indices of $W_x(A_n)$ is zero. We also show that the index of some composed convolution-like operators is zero. Results on the convergence of the pseudospectrum, norms of inverses and condition numbers are also obtained. This talk is based on joint work with B. Silbermann.

In 1912 Hermann Weyl posed the following problem: to characterize the possible sets of eigenvalues of the sum of two Hermitian matrices in terms of the sets eigenvalues of given matrices. In 1962 Alfred Horn recasted this problem as a conjectured series of inequalities for the eigenvalues of the sum matrix. The final step in proving this conjecture was made by A. Klyachko, A. Knutson and T. Tao. We consider the following generalization of Weyl's problem: given the finite sets of eigenvalues of hermitian operators, determine all possible scalar operators that can appear as a sum of these operators. This problem could be reformulated in terms of the existence of the *-representations of certain *-algebras connected with star-shaped graphs. We will also discuss some basic properties of such *-algebras related to extended Dynkin graphs.

We study *-representations of semilinear relations with polynomial characteristic functions. For any finite simple non-oriented graph $\Gamma$ we construct a polynomial characteristic function such that $\Gamma$ is its graph. A full description of graphs which satisfy polynomial (degree one and two) semilinear relations is obtained. We introduce the $G$-orthoscalarity condition and prove that any semilinear relation with quadratic characteristic function and condition of $G$-orthoscalarity is not *-wilde. This class of relations contains, in particular, *-representations of \(U_q(so(3))\).

Variable-coefficient Toeplitz matrices are generated by smooth functions defined on some compact cylinder. Familiar Toeplitz matrices with continuous generating function (defined on the complex unit circle) actually form a very particular class of such matrices and one may ask what asymptotic properties of Toeplitz matrices are valid in this more general context. To this end we describe the structure of the C*-algebra generated by variable-coefficient Toeplitz matrices and study a few spectral quantities (Szegö-like theorems, stability, pseudospectra). The talk is based on joint work with Olga Zabroda and extends earlier results of Kac/Murdock/Szegö and Zabroda/Simonenko.