Abstracts

A linear operator on a Hilbert space $\mathbb{H}$, in the classical approach of von Neumann,  must be symmetric to guarantee self-adjointness. However, it can be shown that the symmetry could be ommited by using a criterion for the graph of the operator and the adjoint of the graph. Namely, $S$ is shown to be densely defined and closed if and only if $\{k+l:\{k,l\}\in G(S)\cap G(S)^*  \}=\mathbb{H}$.

In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint. 

I take the opportunity of Yuri Karlovich’s 70th birthday to indulge in reminiscences of the years between 1993 and 1996, when Yuri was a visiting professor in Chemnitz. We then planned to write a book on non-local operators but soon changed our minds and decided to embark on another topic: the spectral theory of Toeplitz operators with piecewise continuous symbols on Carleson curves with Muckenhoupt weights. I will show you some pictures (both photos and transparencies) from those days and tell you a few pieces of the mathematics we explored in those years.

The Sylvester equation will be analyzed in the sense of its solvability and classification of its solutions. We will show that all solutions are obtained this way, thus creating the general solution. Size of the given set is analyzed with the help of a special operator algebra, which links together some basic operator equations, such as $AX=C$, $XB=C$, $AXB=C$, $X-AXB=C$ and $AX-XB=C$.

Mellin convolution equations acting in Bessel potential spaces are considered. The study is based upon two results. The first one concerns the interaction of Mellin convolutions and Bessel potential operators (BPOs). In contrast to the Fourier convolutions, BPOs and Mellin convolutions do not commute and we derive an explicit formula for the corresponding commutator in the case of Mellin convolutions with meromorphic symbols. These results are used in the lifting of the Mellin convolution operators acting on Bessel potential spaces up to operators on Lebesgue spaces. The operators arising belong to an algebra generated by Mellin and Fourier convolutions acting on $\mathbb{L}_p$-spaces. Fredholm conditions and index formulae for such operators have been obtained earlier by one of the authors and are employed here. The results of the present work have numerous applications in boundary value problems for partial differential equations, in particular, for equations in domains with angular points.

Let $A=(a_{j,k})_{j,k=-\infty}^\infty$ be a bounded linear operator on $l^2(\mathbb{Z})$ whose diagonals $D_n(A)=(a_{j,j-n})_{j=-\infty}^\infty \in l^\infty(\mathbb{Z})$ are almost periodic sequences. For certain classes of such operators and under certain conditions, we are going to determine the asymptotics of the determinants $\det A_{n_1,n_2}$ of the finite sections $A_{n_1,n_2}=(a_{j,k})_{j,k=n_1}^{n_2-1}$ as their size $n_2-n_1$ tends to infinity. Examples of such operators include block Toeplitz operators and the almost Mathieu operator.

The conditions referred to above are rather peculiar and could perhaps be best characterized as smoothness, regularity and diophantine assumptions. The method of the proof consists of a particularly devised Banach algebra approach.

This talk is based on joint work with my student Zheng Zhou and it generalizes results previously obtained together with S.Roch and B.Silbermann.

In recent years, tools such as the Allan-Douglas Local Principle and its generalizations such as the Lifting Theorems and the Local-Trajectory Method have been widely used in the study of the invertibility in non-commutative Banach algebras of functional operators or singular integral operators with shifts.

In this talk we will recall the Local-Trajectory Method, and its generalization using spectral measures, and we will show how it allows us to construct invertibility criteria for operators in a $C^*$-algebra \[{\rm alg}(PQC,U_G)\subset{\mathcal B}(L^2(\mathbb{T}))\] generated by all functional operators of the form $\sum_{g\in F}a_gU_g$, where $a_gI$ are multiplication operators by piecewise quasicontinuous functions $a_g\in PQC$ on $\mathbb{T}$, $U_g:\varphi\mapsto|g'|^{1/2} (\varphi\circ g)$ are unitary weighted shift operators on $L^2(\mathbb{T})$, and $F$ is any finite subset of group $G$, an amenable discrete group of orientation-preserving piecewise smooth homeomorphisms on the unit circle $\mathbb{T}$, which acts topologically freely on $\mathbb{T}\setminus\Lambda^\circ$, where $\Lambda^\circ$ is the interior of a nonempty closed set $\Lambda\subset\mathbb{T}$ composed by all common fixed points for all shifts $g\in G$.

This talk is based on the joint work with M. Amélia Bastos and Yu. I. Karlovich.

The trace-class property of Hankel operators (and their derivatives with respect to the parameter) with strongly oscillating symbol is studied. The approach used is based on Peller’s criterion for the trace-class property of Hankel operators and on the precise analysis of the arising triple integral using the saddle-point method. Apparently, the obtained results are optimal. They are used to study the Cauchy problem for the Korteweg–de Vries equation. Namely, a connection between the smoothness of the solution and the rate of decrease of the initial data at positive infinity is established.

In this talk we consider the scale of Fock spaces $\{F^2_t\}_{t\gt 0}$, where $t \sim \hbar$ acts as a weight parameter and we are interested in the semi-classical limit $t \to 0$. For instance, let $\{T_f^{(t)}\}_{t\gt 0}$ and $\{T_g^{(t)}\}_{t \gt 0}$ be two families of Toeplitz operators acting on $\{F^2_t\}_{t \gt 0}$. What can we say about the product $T_f^{(t)}T_g^{(t)}$ as $t \to 0$? As the term ‘semi-classical’ suggests, we would expect \[\lim\limits_{t \to 0} \|T_f^{(t)}T_g^{(t)} - T_{fg}^{(t)}\|_t = 0\] for reasonable symbols $f$ and $g$. In fact, again for reasonable symbols, the following asymptotic expansion holds: \[T_f^{(t)}T_g^{(t)} = \sum\limits_{j = 0}^{\infty} \frac{(-t)^j}{j!}T_{(\partial^jf)(\bar{\partial}^jg)}^{(t)}\] as $t \to 0$. Note that the right-hand side is not necessarily convergent in the usual sense, but should be interpreted as \[T_f^{(t)}T_g^{(t)} = \sum\limits_{j = 0}^k \frac{(-t)^j}{j!}T_{(\partial^jf)(\bar{\partial}^jg)}^{(t)} + o(t^k)\] for all $k \geq 0$. The aim of this talk is to describe reasonable and unreasonable symbols.

Based on joint work with Wolfram Bauer and Lewis Coburn.

The  $C^*$-subalgebra $\mathfrak B$ of all bounded linear operators on the space $L^2(\mathbb R)$, which is generated by all multiplication operators by piecewise slowly oscillating functions, by all convolution operators with piecewise slowly oscillating symbols and by the range of a unitary representation of the group of all affine mappings on $\mathbb R$, is studied. A faithful representation of the quotient $C^*$-algebra $\mathfrak B^\pi=\mathfrak B/\mathcal K$ in a Hilbert space, where $\mathcal K$ is the ideal of compact operators on $L^2(\mathbb R)$, is constructed by applying an appropriate spectral measure decompositions, a local-trajectory method, a lifting theorem and the Fredholm symbol calculus for the $C^*$-algebra of convolution type operators without shifts. This gives a Fredholm symbol calculus for the $C^*$-algebra $\mathfrak B$ and a Fredholm criterion for the operators $B\in\mathfrak B$.

Suppose $1 \lt p \lt \infty, 1/p \lt s \lt 1+1/p$ and $0 \lt \alpha \lt \tfrac{1}{2}$. Let $A$ be a pseudo-differential operator of order $2 \alpha$, defined to be the generator of a Lévy process. Then, motivated by the notion of the regional fractional Laplacian, we consider the invertibility of the operator $\mathcal{A}_G : H^s_p(\overline{G}) \to H^{s - 2\alpha}_p(\overline{G})$, \begin{equation*}\mathcal{A}_G := r_G A e_G + r_G (A \mathbf{1}_{\mathbb{R}^n \setminus G} ) I, \end{equation*} where $r_G, e_G$ denote the restriction and extension (by zero) operators to the open set $G \subset \mathbb{R}^n$. ($\mathbf{1}_H$ denotes the characteristic function of the set $H$.)

In the special case that $n=1, G= \mathbb{R}_+$ and the symbol $A(\xi) = (1+ \xi^2)^\alpha$, we show that $\mathcal{A}_G$ has a trivial kernel. Moreover, after order-reduction, invertibility can be reformulated in the context of an algebra of multiplication, Wiener-Hopf and Mellin operators. Careful analysis of the resulting symbol shows that $\mathcal{A}_G$ is also Fredholm with index zero. Finally, we briefly review various useful extensions of these results.

Based on joint work with Eugene Shargorodsky.

In two-dimensional elasticity theory there often arise Cauchy singular integral equations, where beside the Cauchy kernel also additional Mellin kernels occur. The invertibility of certain limit operators is necessary and sufficient for the stability of corresponding collocation-quadrature methods. In the set of these limit operators we find two operators which, under certain conditions, belong to the algebra generated by Toeplitz operators with piecewise continuous symbols on the unit circle. We formulate these conditions and present some examples for the application of the mentioned stability theory.

We obtain a sufficient condition for the seminoetherity of singular integral operators with ergodic and piecewise-smooth shift on a circle.

On the Hilbert space $\widetilde{L}_{2}(\mathbb{T})$ the singular integral operator with non-Carleman shift and conjugation $K=P_{+}+(aI+AC)P_{-}$ is considered, where $P_{\pm}$ are the Cauchy projectors, $A=\sum\limits_{j=0}^{m}a_{j}U^{j}$, $a,a_{j}$, $j=\overline{1,m} $, are continuous functions on the unit circle $\mathbb{T}$, $U$ is the shift operator and $C$ is the operator of complex conjugation. We obtain some estimates for the dimension of the kernel of the operator $K$.

During the talk we will discuss compactness of Hankel operators acting between distinct Hardy type spaces. Then, application will be presented to the description of compact commutants and semicommutators of Toeplitz and Hankel operators. The talk is based on a joint work with Karol Leśnik from Poznań University of Technology.

We determine the lower and upper estimates for the essential norm of substitution vector-valued integral operators on Orlicz spaces under certain conditions.

The Jacobi–Trudi formulas imply that the minors of the banded Toeplitz matrices can be written as certain skew Schur polynomials. In 2012, Alexandersson expressed the corresponding skew partitions in terms of the indices of the struck-out rows and columns. In this tlak, we develop the same idea and obtain some new applications. First, we prove a slight generalization and modification of Alexandersson’s formula. Then, we deduce corollaries about the cofactors and eigenvectors of banded Toeplitz matrices, and give new simple proofs to the corresponding formulas published by Trench in 1985.

It is well known that the classical Korovkin and Feller limit theorems are equivalent. During the last four decades intense research has been going on in various generalisations of the celebrated Korovkin's theory especially in the operator algebra settings [1, 2, 3]. This talk is about the study of a similar equivalence of Korovkin & Feller type limit theorems in the operator algebra settings.

References

1. B. V. Limaye and M. N. N. Namboodiri. Weak Korovkin approximation by completely positive linear maps on β(H). J. Approx. Theory, 42(3):201–211, 1984.
2. B. V. Limaye and M. N. N. Namboodiri. A generalized noncommutative Korovkin theorem and ∗-closedness of certain sets of convergence. Illinois J. Math., 28(2):267–280, 1984.
3. MNN Namboodiri, Developments In Non Commutative Korovkin Theorems, RIMS Kokyuroku Series [ISSN1880-2818] 1737-Non Commutative Structure Operator Theory and its Applications, October 27-29, 2010, April 2011.

Let $B(X,Y)$ be the set of all bounded linear operators between Banach spaces $X$ and $Y$. For any subset $\mathcal E\subseteq B(X,Y)$, let $\operatorname{ref}\mathcal E = \{T\in B(X,Y): Tx\in [\mathcal Ex], \forall x\in X\},$ where $[\;]$ denotes the norm closure. We say $\mathcal E$ is reflexive if $\operatorname{ref}\mathcal E = \mathcal E$. Reflexivity of subspaces (unbounded sets) of operators have been extensively studied, in this talk I will discuss results about reflexivity of bounded sets of operators, which is naturally related to the study of operator inequalities and complete positivity of elementary operators.

We consider an eigenvalue problem for a generic one-dimensional convolution integral operator on a finite interval where the kernel is given by a real-valued even function with very mild decay and regularity assumptions. We show how this spectral problem can be solved by two different asymptotic technique which take advantage of the size of the integration interval. Each of these techniques reduces a problem to an integral equation of solvable class and eventually to an ODE. In case of a small interval, this is possible due to existence of a commuting differential operator related to prolate spheroidal harmonics, or alternatively, due to constructive approximation methods for meromorphic Wiener-Hopf matrix factorization. In case of a large interval, the problem, after a sequence of transformations, culminates in an elementary solvable non-homogeneous ODE with boundary conditions obtained via approximate Wiener-Hopf solution of an auxiliary integro-differential equation on a shifted half-line with a Toeplitz plus Hankel kernel. We note that, unlike a finite-rank approximation of the compact operator, both of the auxiliary problems here admit infinitely many solutions (eigenfunctions) as it is the case for the original problem, even though the approximation deteriorates for higher-order eigenvalues/eigenfunctions. These results are a generalization of the author’s previous work (in collaboration with L. Baratchart and J. Leblond) on a convolution equation with Love/Lieb-Liniger/Gaudin kernel.

Let $\Gamma$ be a graph embedded in $\mathbb{R}^{n}.$ We assume that the graph $\Gamma$ is periodic with respect to the lattice $\mathbb{G}$ of dimension $m,1\leq m\leq n$ which acts on $\Gamma$ by shifts $V_{g}u(x)=u(x-g),x\in \Gamma,g\in\mathbb{G}.$ In the natural way we define the spaces $L^{p}(\Gamma),1\leq p\leq\infty$ and let $\mathcal{B}(L^{p}(\Gamma))$ be the Banach algebra of all bounded linear operators acting in $L^{p}(\Gamma).$ We say that a bounded in $L^{p}(\Gamma)$ operator $A$ is band-dominated if for every bounded uniformly continuous function $\varphi$ on $\mathbb{R}^{n}$ \begin{equation} \lim_{t\rightarrow0}\left\Vert \left[  A,\widehat{\varphi_{t}}I\right] \right\Vert _{\mathcal{B}(L^{p}(\Gamma))}=\lim_{R\rightarrow0}\left\Vert A\widehat{\varphi_{t}}I \widehat{\varphi_{t}}A\right\Vert _{\mathcal{B} (L^{p}(\Gamma))}=0,\label{2.1} \end{equation} where $\varphi_t(x)=\varphi(t_1 x_{1},\dots,t_{n}x_{n})$, $t_{j}\in\mathbb{R}$ and $\widehat{\varphi_{t}}=\varphi_{t}\mid_{\Gamma}.$ We denote by $\mathcal{A}_{p}(\Gamma)$ the Banach algebra of all band-dominated operators on $\Gamma.$

Let a sequence $\mathbb{G}\ni h_{k}\rightarrow\infty$ and $A\in\mathcal{B} (L^{p}(\Gamma)).$ We say that the operator $A^{h}\in\mathcal{B}(L^{p} (\Gamma))$ is a limit operator of $A$ if for every $R\gt 0$  \[\lim_{k\rightarrow\infty}\left\Vert P_{R}\left(  V_{-h_{k}}AV_{h_{m_{k}}}-A^{h}\right)  \right\Vert _{\mathcal{B}(L^{p}(\Gamma))}=\lim_{k\rightarrow \infty}\left\Vert \left(  V_{-h_{k}}AV_{h_{k}}-A^{h}\right)  P_{R}\right\Vert_{\mathcal{B}(L^{p}(\Gamma))}=0 \] where $P_{R}$ is an operator of multiplication by the characteristic function of the set $\Gamma\cap\left\{  x\in\mathbb{R}^{n}:\left\vert x\right\vert \lt R\right\}  .$ We say that an operator $A\in\mathcal{B}(L^{p}(\Gamma))$ is rich if every sequence $\mathbb{Z}^{m}\ni g_{k}\rightarrow\infty$ has a subsequence $h_{k}$ defining a limit operator $A^{h}.$ An operator $A\in\mathcal{B}(L^{p}(\Gamma))$ is called locally invertible at infinity if there exists $R\gt 0$ and operators $\mathcal{L}_{R},\mathcal{R}_{R} \in\mathcal{B}(L^{p}(\Gamma))$ such that \[ \mathcal{L}_{R}AQ_{R}=Q_{R},Q_{R}A\mathcal{R}_{R}=Q_{R},\text{where }Q_{R}=I-P_{R}.\]

Theorem 1 Let $A\in\mathcal{A}_{p}(\Gamma),1\lt p\lt \infty$ and $A$ is rich. Then $A$ is locally invertible at infinity if and only if all limit operators $A^{h}$ of $A$ are invertible.

The proof of Theorem 1 is based on results of the book [1], the paper [2], see also [3].

We give applications of this theorem to the investigation of the Fredholm property of wide classes of differential and pseudo-differential operators on periodic graphs.

References

1. Rabinovich, V.S., Roch, S., Silbermann, B.: Limit operators and its applications in the operator theory. In: Operator Theory: Advances and Applications,Vol. 150. ISBN 3-7643-7081-5. Birkhäuser Verlag (2004).
2. Lindner, M., Seidel, M.: An affirmative answer to a core issue onlimit operators. J. Funct. Anal. 267(3), 901-917 (2014).
3. Rabinovich, V.: On the Essential Spectrum of Quantum Graphs, Integr. Equ. Oper. Theory, DOI 10.1007/s00020-017-2386-6, (2017).

The classical Korovkin theorem due to P. P. Korovkin [1] states the following. Let $\{\Phi_{n}\}$ be a sequence of positive linear maps on $C[0,1]$. If $ \Phi_{n}(f)\rightarrow f$ for every $f\in \{1,x,x^{2}\}$, then $ \Phi_{n}(f)\rightarrow f$ for every $f \in C[0,1]$.

Here the convergence is the uniform convergence of sequence of functions. There are several variations of this result into various settings such as Banach algebras, $C^{*}$-algebras etc. In 1999, Stefano Serra Cappizano obtained some Korovkin-type results for obtaining optimal Preconditioners to solve linear systems with Toeplitz structure [2]. The result is for self-adjoint Toeplitz matrices of growing order (they are truncations of Toeplitz operators on the Hardy space of unit circle) which is generated using the Fourier coefficients of a single real valued function $ f \in L^{\infty} (\mathbb{T})$. The test set was the set of all trigonometric polynomials and the notion of convergence is in the sense of eigenvalue clustering of sequences of Hermitian matrices of growing order. Assuming the convergence for trigonometric polynomials, he could prove that the convergence holds for Toeplitz Operators with symbols from the $C^{*}$-algebra of functions generated by the test set.

In [3], these notions were generalized into the setting of operators acting on infinite dimensional Hilbert spaces. It generalized and improved the existing results. Recently, we extended these results to the non self-adjoint Toeplitz matrices [4]. Presently, we consider similar results for Toeplitz operators on Bergman Spaces, Fock spaces etc. We have obtained some similar results for Toeplitz operators on Bergman space with symbols continuous on closed disk. We could prove that the convergence will hold for the Toeplitz Operators with symbols from $C^{*}$- algebra generated by the test set and also for the Operators in the Operator $C^{*}$-algebra generated by the Toeplitz Operators with the symbols from the test set. We try to extend these results to Toeplitz operators corresponding to a large class of symbols containing the continuous symbols.

In the talk, we briefly present these developments. This is a joint work with Prof. Wolfram Bauer, Institute of Analysis, Leibniz University of Hannover, Germany.

References

1. Korovkin, P. P. Linear operators and approximation theory. (1960).
2. Serra-Capizzano, S. A Korovkin-type theory for finite Toeplitz operators via matrix algebras. Numer. Math. 82 (1999), no. 1, 117-142.
3. Kiran Kumar, V. B; Namboodiri, M. N. N.; Serra-Capizzano, S.; Preconditioners and Korovkin-type theorems for infinite-dimensional bounded linear operators via completely positive maps. Studia Math. 218 (2013), no. 2, 95-118.
4. Kiran Kumar, V. B.; Namboodiri, M. N. N.; Rajan, Rahul. A Korovkin-type theory for non-self-adjoint Toeplitz operators; Linear Algebra Appl. 543 (2018), 140-161.

In this research, we introduce several factorizations for the two infinite Hilbert and Cesaro matrices. Firstly we factorize Hilbert matrix based on some well-known matrices, where all these factorization are derived from the Cesaro matrix. One of the results of these factorization is the generalization of Hilbert's inequality. Secondly, we introduce a factorization for Cesaro matrix, which the result is the generalization of Hardy's inequality.

We discuss one to one correspondence between Toeplitz matrices and discrete self-adjoint Dirac systems, introduce new Verblunsky-type coefficients and present Verblunsky-type theorems. Next, we consider Toeplitz-block Toeplitz (TBT) matrices, recover inverse to TBT matrices from the minimal information and describe the structure of the inverse to TBT matrices. We consider also the corresponding reflection coefficients. The talk is based on the papers [1, 2].

References

1. A. L. Sakhnovich, New “Verblunsky-type” coefficients of block Toeplitz and Hankel matrices and of corresponding Dirac and canonical systems. J. Approx. Theory 237 (2019), 186-209.
2. A. L. Sakhnovich, On the structure of the inverse to Toeplitz-block Toeplitz matrices and of the corresponding polynomial reflection coefficients, arXiv:1704.02267 (Transactions Amer. Math. Soc. to appear).

When computing eigenvalues of large matrices the standard implementations are clearly affected and restricted by the underlying finite machine precision. The typical iterative algorithms are more or less susceptible to accumulate small imprecisions resulting in incorrectly computed eigenvalues. Surprisingly, for Toeplitz or Toeplitz-like matrices these resulting errors are often far away from being "small random errors" but appear to be rather systematic with particular attractors. Böttcher, Silbermann and Böttcher, Grudsky described this e.g. in [1, 2].

The aim of this talk is to sketch a theoretical approach which gives insight and an explanation for this phenomenon. This approach is based on operator theoretical and algebraic methods describing the approximation of spectral properties (norms, condition numbers, spectra, and pseudospectra) for band-dominated and similar operators and their truncations.

References

1. A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer-Verlag, New York, 1999.
2. A. Böttcher, S. M. Grudsky, Spectral properties of banded Toeplitz matrices, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.

We define Schur decomposition of a bilinear operator $T: H \times H \rightarrow H$, where $H$ is a Hilbert space. We prove that if $T$ is compact, selfadjoint, and its eigenvalues are ordered, then $T$ has a Schur decomposition. The hypothesis of ordered eigenvalues is fundamental. 

Let $\mathcal{A}^2_{\lambda}(\mathbb{B}^n)$, $\lambda \in (-1,\infty)$, be the standard weighted Bergman space on the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$. And let $T^{\lambda}_a$ denote  the Toeplitz operator, with symbol $a \in L_{\infty}(\mathbb{B}^n)$, acting on $\mathcal{A}^2_{\lambda}(\mathbb{B}^n)$. Given an (Abelian) subgroup $G$ of biholomorphisms of $\mathbb{B}^n$, we are looking for a Bargmann type unitary transform of $\mathcal{A}^2_{\lambda}(\mathbb{B}^n)$ onto a direct sum (or direct integral) of Hilbert spaces \begin{equation*} R_G \, : \ \mathcal{A}^2_{\lambda}(\mathbb{B}^n) \ \longrightarrow \ \bigoplus_{\alpha} H_{\alpha}  \ \ \left( \int^{\oplus}_{\alpha} H_{\alpha} d\alpha \right) , \end{equation*} and a set of $G$-invariant symbols \begin{equation*} S_G = \{ a \in L_{\infty}(\mathbb{B}^n)\, : \ a = a\circ g, \ \mathrm{ for\  all } \ g \in G\}, \end{equation*} such that, for each $a \in S_G$, the operator $R_GT^{\lambda}_aR_G^*$ leaves invariant each Hilbert space $H_{\alpha}$ in the above direct sum (direct integral) decomposition.

In that way a problem of the characterization of the algebra generated by Toeplitz operators $T^{\lambda}_a$, with $a \in S_G$, reduces to the description of the algebras generated by the operators $R_GT^{\lambda}_aR_G^*|_{H_{\alpha}}$ in each of the spaces $H_{\alpha}$. Furthermore certain quantization effects (caused by $\alpha$ tending to infinity) have to be taken into account.

In the talk we present several examples of the above approach characterizing various commutative $C^*$, commutative Banach, and non commutative $C^*$-algebras generated by Toeplitz operators for the case of the two-dimensional unit ball.

The talk is based on a joint work with Wolfram Bauer.