Functional Analysis and Applications Seminar 

The theory of row contractions (called also multicontractions) as initiated by Gelu Popescu is the most notable achievement in the aim of extending the well known Sz.-Nagy-Foias dilation theory of contractions on a Hilbert space. Two objects related to a multicontraction play a significant role in this theory: the Poisson kernel and the characteristic function. The Poisson kernel is an important tool used by Popescu in order to prove the von Neumann inequality for row contractions; the characteristic function is essential in the model theory of completely noncoisometric multicontractions. We give the behaviour of this two objects with respect to the action of the group of unitarily implemented automorphisms of the algebra generated by creation operators on the Fock space.

The recent reflexivity, transitivity and hyperreflexivity results for subspaces and algebras of operators will be presented. We start with the situation when underlying Hilbert space is finite dimensional and giving some examples show that even in this case the notion of reflexivity is interesting. It will be presented that reflexivity and hypereflexivity are equivalent for finite dimensional subspaces of operators even the underlying Hilbert space is not finite dimensional (positive answer for Larson-Kraus problem). We will study the dichotomic behavior (reflexivity versus transitivity) of subspaces of Toeplitz operators on the Hardy space. The Toeplitz operators on the Bergman space will be also considered. We discuss also algebras generated by isometries, power partial isometries and quasinormal operators. The multivariable case will be also presented.

Classes of problems of wave diffraction by a plane angular screen occupying an infinite 270 degrees wedge sector are studied in a Bessel potential spaces framework. The problems are subjected to different possible combinations of boundary conditions on the faces of the wedge. Namely, under consideration there will be boundary conditions of Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, impedance-Dirichlet, and impedance-Neumann types. Existence and uniqueness results are proved for all these cases in the weak formulation. In addition, the solutions are provided within the spaces in consideration, and higher regularity of solutions are also obtained in a scale of Bessel potential spaces.

Partial differential equations on Riemannian manifolds are usually written in intrinsic coordinates, involving metric tensor and Christoffel symbols. But if we deal with a hypersurface, the cartesian coordinates of the ambient space can be applied. 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. This enables, for example, more transparent proofs of Korn's inequalities, tightly connected with solvability and uniqueness of some boundary value problems. Moreover, based on the principle that the displacement minimizes the total free elastic energy at equilibrium, was derived the Lamé operator on the surface (R. Duduchava, D. Mitrea and M. Mitrea). The equation is represented in terms of Günter's derivatives. The Killing's vector fields, solutions of the homogeneous Lamé equation are investigated. Relatively simple form of operators in terms of Günter's and Stoke's derivatives enable simplified treatment of corresponding boundary value problems with the Lax-Milgram lemma and Korn's inequalities with and without boundary conditions. Another approach, the potential method, using fundamental solutions, potential operators, Green formulae and boundary integral equations, available in explicit form, is developed as well. A special accent is made on a thin flexural shell problems in elasticity. We suggest for their study the approach applied to the derivation of Lamé equation. The approach differs from the ones proposed before for modeling linearly elastic flexural shells, suggested by Cosserats (1909), Goldenveiser (1961), Naghdi (1963), Vekua (1965), Novozhilov (1970), Koiter (1970) and many others.

We consider generalized potential operators on a bounded measure metric space with doubling measure satisfying the upper growth condition. Under some natural assumptions on the kernel of the potential, in terms of its almost monotonicity, we present a Sobolev type theorem on the boundedness of such potential operators from the variable exponent Lebesgue space into a certain Musielak-Orlicz space with the N-function defined by the exponent p(x) and the kernel. A reformulation of the obtained result in terms of the Matuszewska-Orlicz indices of the function kernel is also given. The talk is based on a joint paper with M. Hajiboyev (Baku).

We establish higher order asymptotic formulas for traces of Toeplitz matrices with symbols in some generalized Krein algebras. Bounded Wiener-Hopf factorization in decomposing algebras which may contain discontinuous functions plays an essential role in our proofs.

A family of classes $C(Q,R)$ which include all $2\times 2$ factorable matrices with Hölder continuous entries in the unit circle is defined. It is shown that an appropriate characterization of such classes enables one to associate, in a unique way, an algebraic curve to each class $C(Q,R)$ and to show the equivalence of a Riemann-Hilbert problem with matrix coefficient in that class to a scalar Riemann-Hilbert problem in a Riemann surface. This is joint work with A. F. dos Santos and Pedro F. dos Santos.

We will present explicit representations of Bergman kind projections in terms of the Calderon-Zygmund operators on the unit disk $S$ and $S^\ast$. This will permit to decompose the $L^2$ space of unit disk with area measure, in orthogonal spaces of analytical type (or of anti-analytical type). The mentioned Calderon-Zygmund operators will play the role of unitary operators between such analytical type spaces. The dependence on boundary regularity of equalities between Bergman kind projections and singular integral operators, is far way of being understood. We will present easy examples of domains that do not admit Dzhuraev's formulas, exemplify how inside variations of the domain can bring some light to the problem and establish explicit forms of the Bergman and anti-Bergman projections for some open sectors. On the other hand, inside approximation of a domain also permits to get formulas for the poly kernel functions of the upper half plane, to establish corresponding new proofs of Dzhuraev's formulas and to clarify its existence for open sectors. By simple remarks we will achieve known and important properties of Calderon-Zygmund operators $S_j$ for integer $j$. The talk is partially based on joint work with Yu. I. Karlovich.

Positive semi-definite (PSD) operator $A$ "spectrally dominates" PSD operator $B$ if $A^t-B^t$ is PSD for all $t \gt 0$. We (i) give a new characterization of spectral dominance in finite dimensions in terms of a monotonic chain of intermediate, pairwise commuting operators and (ii) determine for which pairs $(A;B)$ spectral dominance persists under the taking of arbitrary compressions. Earlier results about spectral dominance are proven (in finite dimensions) in new ways, and several corollary observations are made. This is a joint work with Charlie Johnson and Bich Hoai, accepted for publication by Proceedings of the AMS (and is accessible for students).

A Fredholm criterion and an index formula are established for Wiener-Hopf operators with matrix semi-almost periodic symbols on weighted Lebesgue spaces for a subclass of Muckenhoupt weights. For matrix almost periodic symbols of Wiener type, an invertibility criterion for Wiener-Hopf operators on weighted Lebesgue spaces is also obtained. These results extend to convolution type operators on weighted Lebesgue spaces on a union of intervals. The talk is based on a joint work with Juan Loreto Hernández.

In many physical and engineering applications, corner singularities of the solutions of elliptic boundary value problems contain important information. One can therefore expect that these singularities have been approximated by the finite element method for a long time. Classical approaches include the augmentation of the finite element spaces by singular functions (Fix method) and the use of dual singular functions for extracting the coefficient of the singular function either directly from the given data or by a post-processing procedure from a computed approximation of the solution. In 2D problems, these methods work well, because the singular functions are all known more or less explicitly, even for the most general elliptic boundary value problems, and there remains only a finite number of coefficients to calculate. In 3D problems, these methods have been studied theoretically, too, and they work well for the case of conical corner singularities. For the case of edge singularities, however, one has the curious situation that several numerical methods have been described and analyzed in detail, up to precise stability and error estimates, but very few actual numerical codes have been implemented. This has two main causes: The unknown coefficients now are functions, living in an infinite-dimensional function space, and the singular and dual singular functions are often too complicated to be used in practice without simplification, whereas over-simplification leads to insufficient precision. A simplification that does work is the recent "quasi-dual singular function method", developed in collaboration with M. Dauge (Rennes) and Z. Yosibash and N. Omer (Beere-Sheva). Here the singular and dual singular functions are approximated by an asymptotic expansion. In this method, moments of the coefficient function are computed from extrapolation of integrals over cylindrical surfaces neighboring the edge involving a finite-element approximation of the solution. The method has been analyzed theoretically in a rather general setting, and it integrates well as a post-processing tool in higher order finite element codes. Numerical results have been obtained for general second order scalar elliptic boundary value problems and for the system of linear elasticity, even in the anisotropic case.

We consider a class of boundary value problems for the three-dimensional Helmholtz equation that appears in diffraction theory. On the three faces of the octant, which are quadrants, we admit first order boundary conditions with constant coefficients, linear combinations of Dirichlet, Neumann, impedance and/or oblique derivative type. A new variety of surface potentials yields \(3 \times 3\) boundary pseudodifferential operators on the quarter-plane that are equivalent to the operators associated to the boundary value problems in a Sobolev space setting. These operators are analyzed and inverted in particular cases, which gives us the analytical solution of a number of well-posed problems. The talk is based upon common research with Ernst Stephan.

We will consider Toeplitz plus Hankel operators generated by symbols which have $n$ points of standard almost periodic discontinuities, and acting between $L^2(T)$ Lebesgue spaces. Conditions are obtained under which these operators are right-invertible and with infinite kernel dimension, left-invertible and with infinite cokernel dimension or simply not normally solvable. This will be done by employing a certain real functional, looking to the resulting signs on the points of standard almost periodic discontinuities. Examples will be provided to illustrate the obtained results.

The problems of wave diffraction by a plane angular screen occupying an infinite 45 degrees wedge sector with Dirichlet and/or Neumann boundary conditions are studied in Bessel potential spaces. Existence and uniqueness results are proved in such a framework. The solutions are provided for the spaces in consideration, and higher regularity of solutions are also obtained in a scale of Bessel potential spaces. The talk is based on a joint paper with D. Kapanadze.

A generalization of the local trajectory method for C* algebras associated with C* dynamical systems based on the notion of spectral measures is presented. This generalization allows to establish invertibility and Fredholm criteria for new classes of C* algebras with shifts. The C* algebra considered in this talk is a non local algebra generated by multiplication operators by slowly oscillating and piecewise continuous functions and by the range of unitary representation of an amenable group of diffeomorphisms with any non empty set of common fixed points. The talk is related to a joint work with C. Fernandes and Y. Karlovich.

We introduce the Lorentz space \(L_{p,q}\) with variable exponents \(p(t), q(t)\) and prove the boundedness of the maximal, singular integral and potential type operators in these spaces. The main goal is to show that the boundedness of these operators in the spaces \(L_{p,q}\) is possible without the local \(\log\)-condition on the exponents, typical for the variable exponent Lebesgue spaces; instead the exponents \(p(t)\) and \(q(t)\) should only satisfy decay conditions of \(\log\)-type as \(t\) tends to \(0\) and infinity. To prove this, we base ourselves on the recent progress in the problem of the validity of Hardy inequalities in variable exponent Lebesgue spaces.

The talk is based on a joint paper with V. Kokilashvili.

We propose an algebraic approach to the stability problem for the finite sections of general band-dominated operators acting on $l^p = l^p(Z)$ for $p \gt 1$. This approch allows us to get new results which previously were known mainly for $p = 2$. One of the main results shows that a band-dominated operator is Fredholm if and only if the approximation numbers of its finite sections have a special behavior.

We prove a Fredholm criterion for operators in the Banach algebra of singular integral operators with matrix piecewise continuous coefficients acting on a variable Lebesgue space with a radial oscillating weight over a logarithmic Carleson curve. The local spectra of these operators are massive and have a shape of spiralic horns depending on the value of the variable exponent, the spirality indices of the curve, and the Matuszewska-Orlicz indices of the weight at each point. These results extend (partially) the results of A. Böttcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to the case of variable Lebesgue spaces.

Let $E$ be a linear space and $p$ of $E$ into a Riesz space $Y$ a vectorial norm. The space $E$ with $p$ is named a vectorially normed space. Of special interest among vectorially normed spaces are the vectorial inner product spaces, the theory of which is richer and retains many features of Euclidean spaces. In these spaces the vectorial norm $p$ is defined in terms of a vectorial inner product. Here we consider $Y$ to be a $B$-regular Yosida space, that is a unitary Archimedean and Dedekind complete Riesz space such that the intersection of all its hypermaximal bands is the zeroelement. In Yosida Theorem we assert that any $B$-regular Yosida space is Riesz isomorphic to the space $B(A)$ of all bounded real-valued mappings on a certain set $A$. Next we prove Bessel Inequality and Parseval Identity for a vectorial inner product with range in a $B$-regular and norm complete Yosida algebra.

The C*-subalgebra $B$ of bounded linear operators on the $L^2$ space over the unit circle $T$ which is generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular integral operator and by the range of a unitary representation of an amenable group of orientation-preserving diffeomorphisms (shifts) of $T$ onto itself with any nonempty set of common periodic points is studied. A symbol calculus for the C*-algebra $B$ and a Fredholm criterion for its elements are obtained. For the C*-algebra $A$ composed by all functional operators in $B$, an invertibility criterion for its elements is also established. Both the C*-algebras $B$ and $A$ are investigated by using a generalization of the local-trajectory method for C*-algebras associated with C*-dynamical systems which is based on the notion of spectral measure. The results essentially depend on the structure of the set of periodic points of shifts. The talk is related to a joint work with M.A. Bastos and C.A.Fernandes.