Functional Analysis and Applications Seminar 

We prove analogues of the Brown-Halmos and Nehari theorems on the norms of Toeplitz and Hankel operators, respectively, acting on subspaces of Hardy type of reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.

The talk is aimed at the computation of the partial indices of regular matrix functions knowing the values of these functions at distinguished points. The proposed algorithm is not only stable but also converges sufficiently fast provided the underlying function is smooth.

The talk is devoted to calculating the indices of Toeplitz operators with oscillating matrix symbols on the Hardy space over the upper half-plane. The symbols belong to the $C^{*}$-algebra generated by semi-almost periodic and slowly oscillating matrix functions. An approach which does not use the harmonic extensions of slowly oscillating matrix functions is presented.

Further developments of the results on the Kontorovich-Lebedev type transformations and associated convolution in $L_p$ spaces are given. In particular, we consider compositions with the Mellin type convolution transformations and obtain some interesting cases of integral transforms depending upon parameters of hypergeometric functions (the Lommel functions, the Whittaker functions, the Clausenian functions, etc.). Boundedness properties are investigated, Plancherel type theorems are proved. Parseval type identities are established. Applications to integral equations of the Fredholm and convolution type associated with the Kontorovich-Lebedev operator are given.

Usando localização estabelecem-se critérios de Fredholm para a álgebra-$C^\ast$ gerada pelas projecções de Bergman e anti-Bergman com coeficientes seccionalmente contínuos no semi-plano superior. As secções consideradas são definidas por curvas regulares tais que em pontos da fronteira do semi-plano coincidem numa vizinhança com segmentos de recta. A hipótese anterior relaciona-se com a aplicação dum trabalho de Plamenevsky, baseado numa decomposição da transformada de Fourier multidimensional. A estratégia cria a necessidade de estabelecer uma particular caracterização de certas álgebras-$C^\ast$ geradas por projecções ortogonais em espaços de Hilbert. É igualmente estabelecido, usando $\ast$-isomorfismos locais, um $\ast$-isomorfismo entre a álgebra considerada e uma correspondente no disco unitário. O seminário é baseado num trabalho conjunto com Y. Karlovich.

The problems connected with Gaudin models are reviewed by analyzing models related to the trigonometric $\operatorname{osp}(1|2)$ classical $r$-matrix. Moreover, the eigenvectors of the trigonometric $\operatorname{osp}(1|2)$ Gaudin Hamiltonians are found using explicitly constructed creation operators. Commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik-Zamolodchikov equation yields the norm of the eigenvectors. The generalized Knizhnik-Zamolodchikov system is discussed both in the rational and the trigonometric case.

Uma sucessão chama-se sucessão frequencial de Pólya (ou multiplamente positiva) de ordem $r$, se todos os menores de ordem menor ou igual a $r$ (todos os menores se $r$ é igual a $\infty$) da sua matriz de Toeplitz são não negativos. A nossa comunicação estará relacionada com as singularidades de funcões geradoras dessas sucessões (f.g.'s PFr) para $r$ finito. Para cada $r$ finito, descreveremos, em termos de ordens próximas, o possível crescimento que uma f.g. PFr pode ter no seu círculo de convergência e perto das suas singularidades. Serão ainda descritos todos os possíveis domínios de holomorfia de f.g.'s PFr. Para a obtenção dos resultados acima mencionados foram desenvolvidos dois novos métodos de construção de f.g.'s PFr não inteiras.

For the Wiener-Hopf factorization of $2\times 2$ matrix functions defined on a closed Carleson curve, so-called rational transformations, i.e. multiplication by rational matrix functions, are important. In the first part of the lecture, the topic will be motivated by a variety of applications and by general operator theoretical facts as well. Then we establish a classification scheme for $2\times 2$ matrix functions, which is based on such transformations. We determine invariants under these transformations and describe those matrix functions which can be transformed to triangular or Daniele-Khrapkov form. In the third part we consider special rational transformations and study the same problem. For instance, we consider transformations with matrix functions that are analytic and invertible on an open neighborhood of the given curve.

The talk is based upon common work with Torsten Ehrhardt, to be published in Linear Algebra and Its Applications under the same title.

In this talk we deal with a Fredholm theory in the algebra $A$ generated by operators with oscillating symbols and the Cauchy singular operator as a subalgebra of bounded linear operators in ($n$-vector) Lebesgue spaces.

In the first part of the talk, using the limit operator theory, a necessary Fredholmness condition for any operator in $A$ is established and, from one of the local principles, sufficient Fredholmness conditions for some elements in this algebra are obtained. A Fredholm criterion for Toeplitz operators with matrix symbols in the $C^{*}$ algebra $C$ generated by slowly oscillating and semi-almost periodic matrix functions on $ℝ$ is established as a consequence.

In the second part, using the notion of harmonic extension, an index theory for Fredholm Toeplitz operators whose generating functions belong to the $C^{*}$ algebra $C$ is developed. The most relevant result is obtained by reducing to Toeplitz operators whose generating functions belong to the space of semi-almost periodic matrix functions.

This talk is based on common work with Y. Karlovich and B. Silbermann.

We prove the inverse closedness of the Banach algebra $A$ of functional operators with non-Carleman shifts, which have only two fixed points, in the Banach algebra of all bounded linear operators on $L^p$. We suppose that the generators of the algebra $A$ have essentially bounded data. An invertibility criterion for functional operators in $A$ is obtained in terms of the invertibility of a family of discrete operators on $l^p$. An effective invertibility criterion is established for binomial difference operators with bounded coefficients on the spaces $l^p$. Using the reduction to binomial difference operators, we give effective criteria of invertibility for binomial functional operators on the spaces $L^p$.

These results are obtained in collaboration with Yuri Karlovich.

A complex Banach space possesses an intrinsic algebraic structure associated with the holomorphic properties of its open unit ball. This introduction will give describe the algebraic structure in a variety of examples, and will also give examples to show how the holomorphic, geometric and algebraic properties of complex Banach spaces are interwoven.

Last decade there was intensively developed the theory of Lebesgue spaces with variable exponent when the order $p$ of integrability depends on $x$. (These spaces have interesting applications in fluid mechanics). The corresponding theory proved to be difficult to develop because these spaces are not invariant neither with respect to translation nor dilation. It suffices to mention that for example Young type theorems for convolutions are not already valid in these spaces. In general, convolution operators have a "bad" behaviour in such spaces. A progress was recently made by proving the uniform boundedness of dilation convolution operators under some natural assumptions on the kernel of the convolution. This result, presented in particular, in the talk allowed us to prove that "nice" functions (infinitely differentiable with compact support) are dense not only in the Lebesgue spaces with variable exponent, but also in Sobolev spaces generated by them. However, boundedness of the singular integral operators remained an open question for a long time. We show that some modification of the method developed in the above investigation allows us to prove also that the singular operator along a bounded Lyapunov curve is bounded in the space with a variable exponent $p(x)$ under some natural assumptions on $p(x)$. The last topic of the talk is based on the joint research with Prof. Vakhtang Kokilashvili.

Infinite systems of linear equations are usually solved by the method of finite sections. This means that the infinite system is replaced by the sequence of finite sections of the original system and it is expected that the solutions of the finite systems converge to the solution of the infinite system. This method has a rich history of 150 years and many distinguished mathematicians have made important contributions in this area. The talk will present the early history and recent important achievements. Unexpected examples and computational experiments will motivate and illustrate the main results. Special attention will be paid to the case of Toeplitz matrices with continuous and discontinuous symbols.

The Dirichlet and Neumann problems for harmonic functions from the Smirnov type classes in domains with arbitrary piecewise smooth boundaries will be discussed. The picture of solvability is described completely; the non-Fredholm cases are exposed; an influence of geometric properties of boundaries on the solvability is revealed; a criterion for the unique solvability of this problem is established for arbitrary boundary values from the Lebesque spaces with exponent greater than one. Similar problems are considered in weighted Smirnov classes of harmonic functions. The weight is an arbitrary power function. In the classes of harmonic functions which are real parts of analytic functions represented in the domains by the Cauchy type integrals we investigated the Dirichlet problem with boundary functions from the weighted Zygmund classes. In all the cases of solvability there are given explicit formulas for the solution in terms of Cauchy type integrals and conformal mapping functions. The proofs are heavily based on the investigation of the linear conjugation problem with oscillating conjugation coefficient, the boundary properties of derivatives of functions which map conformally the unit circle onto a domain with an arbitrary piecewise smooth boundary and two-weighted norm inequalities for singular integrals. The talk is based on joint papers by V.Kokilashvili, V. Paatashvili and Z.Meshveliani.