Functional Analysis and Applications Seminar 

The numerical values (associated with the numerical ranges) and \(q\)-numerical values (associated with the \(q\)-numerical ranges) of two Hilbert space operators are compared. The main result of the talk states that the absolute values of the \(q\)-numerical value function of two operators coincide if and only if the operators are unimodular scalar multiples of each other, for \(q\) positive and less than one. In the extreme cases when \(q\) is equal to one or to zero, additional possibilities occur. These statements are framed in terms of \(C\)-numerical ranges where the operator \(C\) is nonscalar of rank one. The results are motivated by an application to the problem (still largely unsolved) of characterizing norm preservers of Jordan products of matrices. (Joint work with B. Kuzma, G. Lesnjak, C.-K. Li, T. Petek.)

In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces $L^p(G)$ over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces $L^p(G)$ where $p(\cdot)$ satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.

A \(n\times n\) matrix \(A\) has normal defect one if it is not normal, however can be embedded as a north-western block into a normal matrix of size \((n+1)\times (n+1)\). The latter is called a minimal normal completion of \(A\). A construction of all matrices with normal defect one is given. Also, a simple procedure is presented which allows one to check whether a given matrix has normal defect one, and if this is the case, to construct all its minimal normal completions. A characterization of the generic case for each n under the assumption that the rank of the self-commutator of \(A\) equals \(2\) (which is necessary for \(A\) to have normal defect one) is obtained. Both the complex and the real cases are considered. It is pointed out how these results can be used to solve the minimal commuting completion problem in the classes of pairs of \(n \times n\) Hermitian (resp., symmetric, or symmetric/antisymmetric) matrices when the completed matrices are sought of size \((n+1)\times (n+1)\). An application to the \(2\times n\) separability problem in quantum computing is described. This is a joint work with Dmitry Kaliuzhnyi-Verbovetskyi and Hugo Wourdeman.

The existence and construction of common invariant cones for families of real matrices is considered. The complete results are obtained for \(2\times 2\) matrices (with no additional restrictions) and for families of simultaneously diagonalizable matrices of any size. Families of matrices with a shared dominant eigenvector are considered under some additional conditions. This is a joint work with Leiba Rodman and Hakan Seyalioglu.

Starting from M. Putinar's result showing that a hyponormal operator has a scalar extension which means that it is similar to the restriction to an invariant subspace of a (generalized) scalar operator (in the sense of Colojoara-Foias), we discuss this notion and show that backward Aluthge iterates of hyponormal operators are subscalar.

We study the problem of weighted boundedness of the one-dimensional singular operator $S$ with Cauchy kernel in Morrey spaces on a curve. The weight function may be a product of a finite number of almost monotonic functions with nodes on the curve. The boundedness of the operator $S$ in case of such a weight is reduced to the boundedness of Hardy type operators in Morrey spaces with this weight. We prove the latter, which enables us to obtain sufficient conditions for the boundedness of the singular operator $S$ in terms of the Matuszewska-Orlicz indices of weights. A special attention is paid to the case of power weights where we prove that the conditions on the weight are also necessary for the boundedness. We also discuss an application to the study of Fredholmness of singular integral equations in weighted Morrey spaces, the interest to this investigation being caused by the non-separability of Morrey spaces.

Harold Widom proved in 1966 that the spectrum of a Toeplitz operator $T(a)$ acting on the Hardy space $H^p(T)$ over the unit circle $T$ is a connected subset of the complex plane for every bounded measurable symbol $a$ and $p \gt 1$. In 1972, Ronald Douglas established the connectedness of the essential spectrum of $T(a)$ on $H^2(T)$. We show that, as was suspected, these results remain valid in the setting of Hardy spaces $H^p(G,w), p \gt 1,$ with general Muckenhoupt weights $w$ over arbitrary Carleson curves $G$. This is a joint work with Ilya Spitkovsky.

We will consider initial value problems for functional equations on the half-axis that contain Hilbert transforms, derivatives and complex shifts. The class of problems is motivated by various applications, and will be considered in both Bessel potential and Sobolev–Slobodeckij space settings. Results are Fredholm and invertibility criteria as well as explicit analytical solution in cases where techniques for the constructive factorization of symbol matrices are available. The talk is based on a joint work with R. Duduchava and F.-O. Speck.

A material is called hemitropic (acentric, noncentrosymmetric, chiral) if it is isotropic with respect to orthogonal transformations, but not with respect to mirror reflections. Typical examples are quartz crystals, DNA, bones and nanotubes. Although engineers have studied hemitropic materials since the early 1960s, the theory of elasticity for hemitropic materials has only very recently become the object of rigorous mathematical analysis. The governing equations describing time harmonic elastic fields in hemitropic materials, constitute a system of two vector elliptic PDEs of second order, stated in terms of the displacement and the microrotation vectors. When all the “hemitropicity” parameters become zero, this system reduces to the well-known reduced Navier equation of classical (isotropic) linear elasticity. In this talk, we present the basic boundary value and transmission problems arising in 3-dimensional linear hemitropic elasticity. We construct the fundamental solution that satisfies a Sommerfeld-Kupradze type radiation condition. We establish Green’s formulae, based on which we obtain representations of solutions (in bounded and unbounded domains) in terms of appropriate potentials (single-layer, double-layer, Newtonian). We study the jump and mapping properties of these potentials, and of the corresponding boundary integral (pseudodifferential) operators. Using Potential Theory and the Theory of Pseudodifferential Operators we establish the uniqueness and the existence of solutions (in Hölder, Sobolev and Besov spaces) to the Dirichlet, Neumann, and mixed type boundary value problems and to transmission problems. Further, we study the regularity properties of the solutions to these problems. The solvability is treated in both the cases of smooth and of Lipschitz domains.

I intend to provide as much as the time permits with the basic facts of theory of unbounded subnormal operators. This is a class of operators living somehow aside, out of the main interest of operator theorist, presumably because of lack standard tools. My goal is to convince the audience that this is much undeserved. I hope some distinguised example from quantum mechanics will help me to fulfil the promise.

We develop a pseudodifferential calculus on $\mathbb{R}^n$ suitable to the study of the Fredholm index from the topological viewpoint, namely to generalizations of the index formula on compact spaces. We consider operators that are multiplication by a matrix valued function outside a compact set and give some properties of this class. We then study the closure of a suitable subalgebra. In particular, we find explicit Fredholm criteria and show that the symbols of Fredholm operators have similar topogical properties to those on compact spaces, leading in the same way to a topological formula for the index. Our results have applications on wider classes of pseudodifferential operators on $\mathbb{R}^n$ , namely the isotropic and scattering calculi, and also to pseudodifferential operators on non-compact manifolds.

Sz. Nagy and Foias used an approach based upon the Wold decomposition of an isometry for proving factorization results for operator-valued functions on the unit circle. We are applying an analogue of the Wold decomposition for semigroups of isometries to give some geometric insight into factorization results by Helson and Lowdenslager for matrix-valued functions defined on compact groups with a totally ordered dual. By using some counterexamples, we show that the extensions of these results to operator- valued functions face some basic obstructions. The talk is based on joint work with Dan Timotin (Mathematical Institute of the Romanian Academy).

We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights $w(s)=\left|(s-t)^c\right|$, where $c$ is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point $t$ and $c$ is not real, then the weight $w$ is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.

A well-known theorem of Fong and Sourour states that an elementary operator acting on the space $B(H)$ of all bounded linear operators on a Hilbert space is compact if and only if the symbols of the operator can be chosen to be compact. In this talk we will give quantitative versions of this result using the notions of an \(s\)-number function introduced by A. Pietsch and the theory of ideals of $B(H)$ developed by von Neumann, R. Schatten and W. Calkin. We will relate the behaviour of the $s$-numbers of a given elementary operator to that of its symbols. We will further extend these results to the case of elementary operators acting on general C*-algebras. The talk is based on a joint work with M. Anoussis and V. Felouzis.

There are some reasons to assume that Galerkin's method (or what is called now Galerkin's method) was born about 100 years ago. It is not quite clear to whom one has to adress the priority, but without doubt, Bubnov, Galerkin, Ritz and Simic belong to the circle of main actors. Interestingly enough, the first idea of Galerkin's method was created in order to solve approximately some biharmonic problems occuring in the theory of thin plates. I shall try to describe in short a part of these developments which then had a considerable influence on forming Numerical Mathematics both for biharmonic problems and yet for more general settings. Subsequently I will mention some theoretical concepts of projection and more general approximation methods for solving operator equations and then pass back to biharmonic problems. Especially I will discuss a method of approximate solution of such problems based on function theory considerations.

The definition of compactness (and that of weak compactness) for a linear map between normed spaces may be extended to multilinear maps in a fairly natural way. We treat compactness as a sort of "smallness" condition for multilinear maps. For Banach algebras $A$ and $B$ we call a linear map, $T: A \rightarrow B$ , a cf-homomorphism (meaning "compact from a homomorphism") if the bilinear map $S : A \times A \rightarrow B$ , $S(a,b) = T(a)T(b) - T(a b)$ (i.e. if the "failure to be multiplicative") is a compact bilinear map. We give general theorems showing that such maps are rather well behaved as well as numerous examples. In particular we characterise the pairs of compact, Hausdorff spaces $X$ and $Y$ for which cf-isomorphisms from $C(X)$ to $C(Y)$ are automatically multiplicative.

The talk is devoted to studying the Wiener-Hopf operators with symbols generated by semi-almost periodic and slowly oscillating matrix functions with entries in the Banach algebra of Fourier multipliers on weighted Lebesgue spaces. Fredholm and invertibility results are obtained. The talk is based on a joint work with Juan Loreto Hernández.

The partial order relation existing in the set of tripotents $U(B)$ in a $\mathrm{JBW}$*-triple $B$ is such that it suffices to adjoin a greatest element for the new set to become a complete lattice. For example, such is the case of the partial isometries in a $W$*-algebra which, as it is well-known, are exactly the tripotents in the algebra. Any $\mathrm{JB}$*-subtriple $A$ of $B$ determines a subset $R(A)$ of tripotents called the range tripotents relative to $A$. It is the aim of this talk to present new results concerning the restricted partial order relation in $R(A)$. Furthermore, an analysis of the suprema of families of range tripotents will lead to establishing a counterpart of a result already existing for projections in $W$*-algebras. It is intended to provide the background material needed to make the talk understandable to an audience familiar with operator algebras.

The invertibility of Wiener-Hopf plus Hankel operators with essentially bounded Fourier symbols is characterized via certain factorization properties of the Fourier symbols. In addition, a Fredholm criterion for these operators is also obtained and the dimensions of the kernel and cokernel are described. The talk is based on a joint paper with L. P. Castro.

Let $T$ be a bounded operator on some Hilbert space and $\mathrm{Al}(T)$ be its Aluthge transform. $T$ and $\mathrm{Al}(T)$ share in general their spectral properties and have been intensively treated this last decade. In our talk the cyclic behaviour is investigated. We show that adjoints have he same cyclic behaviour. This is done through some commutation techniques.