Abstracts

This work represents the concept of an $n$-groupoid $ \Gamma^n $ and n-characters $ \chi_n $ on $n$-groupoids as complex-valued maps from spaces of different classes of morphisms satisfying the condition $ \chi_n (\psi \circ_k \varphi) = \chi_n (\psi) + \chi_n (\varphi) $ for any possible compositions. A sequence of spaces of n-characters and morphisms between them is constructed and its accuracy is shown.

This construction has important application for describing the derivations in a group algebras. In particular, this approach allows us to study the algebra of external derivations from a new point of view, and also to construct some interesting examples. The work was carried out under the guidance of Arutyunov A. A.. And it is based on the Mishchenko A. S. ideas.

In this contributed talk, we study one step iterative scheme to approximate common fixed points of two generalized non-expansive mappings  in uniformly convex Banach spaces and using the same scheme we prove some weak and strong convergence results for such mappings. Further, we establish some weak and strong convergence results for a finite family of generalized non-expansive mappings to approximate common fixed points using proposed algorithm in uniformly convex Banach spaces. As an application, we approximate the solution of image recovery problem in Banach space setting. To support our results we illustrate some numerical examples. Our results are new and generalize several relevant results in literature.

In this talk we present some bounds of numerical radius of a bounded linear operator on a complex Hilbert space which improves on the existing bounds. Also we would like to present a Haagerup-Harpe type inequality for numerical radius. As an application we estimate the zeros of a given polynomial.

This work is jointly done with Prof K. Paul and Mr P. Bhunia.

The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz (1982). Previously, J. Lindenstrauss (1966) studied similar versions of this notion in the context of non-separable reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the existence of linear Hahn-Banach extension operators via interspersing subspaces in a purely Banach space theoretic set up. In this paper, we study similar questions in the context of Banach modules and module homomorphisms, in particular, Banach algebras of operators on Banach spaces. Based on Dales, Kania, Kochanek, Kozmider and Laustsen (2013), and also Kania and Laustsen (2017), we give complete answers for reflexive Banach spaces and the non-reflexive space constructed by Kania and Laustsen from the celebrated Argyros-Haydon’s space with few operators.

In this work, the concept of $n$-power-hyponormal operators on a hilbert space defined by Messaoud Guesba and Mostefa Nadir in (M. Guesba and M. Nadir, On operators for which $T^2\geq -T^{*2}$; The Australian Journal of Mathematical Analysis and Applications) is generalized when an additional semi-inner product is considered. This new concept is described by means of oblique projections.

For a a real normed space $(X,\|\cdot\|)$ and for $x,y\in X$, we consider the Birkhoff orthogonality relation (cf. [1]): $$ x\bot_{B}y\quad\Longleftrightarrow\quad \forall\,\lambda\in\mathbb{R}:\quad \|x+\lambda y\|\geq\|x\|. $$ An approximate version of this orthogonality can be defined (cf. [2]) by: $$ x\bot^{\hspace{-0.2em}\varepsilon}_{B}y\quad\Longleftrightarrow\quad \forall\,\lambda\in\mathbb{R}:\quad \|x+\lambda y\|^2\geq\|x\|^2-2\varepsilon\|x\|\,\|\lambda y\| $$ with $\varepsilon\in [0,1)$. Moreover, the following characterisation is in our disposal (cf. [3]): $$ x\bot^{\hspace{-0.2em}\varepsilon}_{B}y\quad\Longleftrightarrow\quad \exists\, z\in \operatorname{Lin}\{x,y\}:\quad x\bot_{B}z,\quad \|z-y\|\leq\varepsilon\|y\|. $$ We will discuss the above notions, further characterizations and its applications, in particular in operator theory. We will also consider the approximate symmetry of the Birkhoff orthogonality, i.e. the property: $$ x\bot_{B} y\quad \Longrightarrow\quad y\bot^{\hspace{-0.2em}\varepsilon}_{B} x,\qquad x,y\in X, $$ as well as its connections with geometrical properties of the considered space (cf. [4]).

References

1. Birkhoff G., Orthogonality in linear metric spaces, Duke Math. J., 1 (1935), 169-172.
2. Chmieliński J., On an $\varepsilon$-Birkhoff orthogonality, J. Inequal. Pure and Appl. Math. 6 (2005), Art. 79.
3. Chmieliński J., Stypuła T., Wójcik P., Approximate orthogonality in normed spaces and its applications, Linear Algebra and its Applications 531 (2017), 305-317.
4. Chmielińnski J., Wójcik P., Approximate symmetry of the Birkhoff orthogonality, J. Math. Anal. Appl. 461 (2018), 625-640.

The mathematical theory of quantum communication science is generally studied in the context of Hilbert spaces. The main resource in such communications is the quantum entanglement. The basic unit of quantum information is qubit which is a member of a Hilbert space. The entangled states belong to tensor products of appropriate Hilbert spaces. This new paradigm in technology was started in the work of Bennet et al (Phys. Rev. Lett. 70(1993) 1895) which is known as the teleportation protocol. In the present context we review the development following the above mentioned work and against that background present a protocol for preparing a three qubit state at a remote location. It is finally accomplished by applications of tensor products of appropriate unitary operators followed by a CNOT operation.

Given two Banach spaces $X$ and $Y$ let $\mathcal{L}(X, Y )$ denote the vector space of operators acting between them; its derived functor is the one that assigns to each couple $X; Y$ the vector space $\operatorname{Ext}(X, Y)$ of exact sequences $0 \to Y \to \Box \to X \to 0$  modulo equivalence; let us agree that the second derived functors will be called $\operatorname{Ext}^2(X, Y)$.

Several important Banach space problems and results adopt the form $\operatorname{Ext}(X, Y) = 0$ (or $\operatorname{Ext}(X, Y) = 0$). For instance,

•  Sobczyk's theorem: $\operatorname{Ext}(c_0, X) = 0$ for every separable Banach space $X$.
•  Lindenstrauss's lifting principle:  $\operatorname{Ext}({L}_1(\mu), X) = 0$ for  every ultrasummand $X$.
•  The Enflo-Lindenstrauss-Pisier and Kalton-Peck construction: $\operatorname{Ext}(\ell_2, \ell_2) \neq 0$.
•  The Johnson-Zippin's theorem: $\operatorname{Ext}(H^*, \mathcal{L}_\infty) = 0$ for every subspace $H$ of $c_0$.

In general, a basic Banach space question is whether $\operatorname{Ext}(X, Y) = 0$ for a given couple of Banach spaces $X$, $Y$. Similar questions for $\operatorname{Ext}^2$ have not been treated. Let us write $\operatorname{Ext}^2(X, Y)=0$ to mean that all elements $FG$ of $\operatorname{Ext}^2(X, Y)$ are $0$.

Palamodov's Problem 6 in [2] says: Is $\operatorname{Ext}^2(\cdot,E)=0$ for any Fréchet space? A solution to Palamodov's problem in the category of Fréchet space was provided by Wengenroth. Let us answer it in the negative even in the domain of Banach spaces.

Perhaps the most interesting situation is the Hilbert space case:  Is $\operatorname{Ext}^2(\ell_2, \ell_2)=0$?

For which a few partial results can be obtained.  The first one establishes an unexpected connection between homology and the study of bilinear forms [1]:

Theorem Let $X$ be a Banach space and let $Q: \ell_1(\Gamma)\to X$ be a quotient map. $\operatorname{Ext}^2(X, X^*)=0$ if and only if every bilinear form defined on $\ker Q$ can be extended to a bilinear form on $\ell_1(\Gamma)$.

The second result connects the $\operatorname{Ext}^2$ problem with the nature of subspaces of $\ell_1$. Precisely,

Theorem Let $X$ be a separable Banach space and let $q: \ell_1\to X$ be a quotient map.

•  If $\ker q$ is an $\mathcal{L}_1$-space then $\operatorname{Ext}^2(X, X^*)= 0$. 
•  If $\ker q $ has an unconditional basis and is not an $\mathcal{L}_1$-space then $\operatorname{Ext}^2(X, X^*) \neq 0$.

Joint work with Jesús M F Castillo (University of  Extremadura). This work was supported by projects MTM2016-76958-C2-1-P and  IB16056 of Junta de Extremadura.

References

1. J. M. F. Castillo and R García, Bilinear forms in the homology of Banach spaces, Linear Algebra and its Applications, 566 (2019)199-211.
2.  V. Palamodov, The projective limit functor in the category of topological linear spaces. (Russian) Mat. Sb. (N.S.) 75 (117) 1968, 567--603 (English Transl. Math-USSR-Sb 4 (1968) 529-558).

Birkhoff-James orthogonality is a generalization of Hilbert space orthogonality to normed spaces. In a given normed space $\mathscr X$, an element $x$ is said to be Birkhoff-James orthogonal to another element $y$ if $$\|x+\lambda y\|\geq \|x\| \text{ for all } \lambda \in \mathbb C.$$ We discuss characterizations for this orthogonality to hold when $\mathscr X$ is a $C^*$-algebra. These characterizations give rise to interesting distance formulas for an element $a$ of $\mathscr X$ to the one dimensional subalgebra $\mathbb C\ b$, generated by an element $b$. More generally, orthogonality to a subspace can be defined and subsequently distance formulas for $a$ to a subspace $\mathscr B$ of $\mathscr X$ can be obtained, when best approximation from $a$ to $\mathscr B$ exists. 

Here we present several necessary and sufficient conditions for the existence of Hermitian positive definite solutions of nonlinear matrix equations of the form $X^s+A^*X^{-t}A+B^*X^{-p}B = Q$, then we use this idea to solve the pair of nonlinear matrix equations of the form \begin{align*} & X^{s_1}+A^*X^{-t_1}A+B^*Y^{-p_1}B=Q_1\\
& Y^{s_2}+A^*Y^{-t_2}A+B^*X^{-p_2}B=Q_2\end{align*} where $s, t, p, s_1, t_1, p_1, s_2, t_2, p_2 \geq 1;~A, B$ are nonsingular matrices and $Q, Q_1, Q_2$ are Hermitian positive definite matrices. Using matrix inequalities, first we give some necessary conditions for the existence of Hermitian positive definite solution. Then we provide sufficient conditions for the existence and uniqueness of solution. Finally, we give some examples.

The metric $d(A,B)=\left[\frac{1}{2}(tr\, A+tr\, B)-tr(A^{1/2}BA^{1/2})^{1/2}\right]^{1/2}$ on the manifold of $n\times n$ positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is of interest in differential geometry, as it is the distance function corresponding to a Riemannian metric. We study some fundamental properties of this metric, and discuss the barycentre of positive definite matrices with respect to it. This is based on joint work with Rajendra Bhatia and Yongdo Lim.

We completely characterize Birkhoff-James orthogonality with respect to numerical radius norm in the space of bounded linear operators on a complex Hilbert space. As applications of the results obtained, we estimate lower bounds of numerical radius for $n\times n$ operator matrices, which improve on and generalize existing lower bounds. We also obtain a better lower bound of numerical radius for an upper triangular operator matrix.

This is a joint work with Professor Kallol Paul and Mr. Jeet Sen, arXiv:1903.06858v1 [math.FA] 16 Mar 2019.

In this talk we consider certain important metrics on the positive definite cones in $C^\ast$-algebras. These are the Thompson part metric and the Bures-Wasserstein metric. We describe the precise structures of the corresponding surjective isometries. More general generalized distance measures, especially certain kinds of quantum relative entropies, are also considered and the related preserver transformations are characterized.

The notion of amenability arose in group theory. Its origin can be driven back to the Banach-Tarski paradox. Later on Barry Johnson proved that a locally compact group $G$ is amenable if and only if its convolution algebra $L^1(G)$ satisfies a certain cohomological property. This gave rise to an abstract definition of the so-called amenable algebra and started a vast development of the theory of amenable Banach algebras. Further research — through functorial language — led Taylor, Johnson and Helemskii to a definition of an amenable topological algebra. In this talk we will focus — after recalling necessary definitions — on a special class of Fréchet algebras, the so-called Köthe echelon algebras. These are sequence spaces of the form \[\lambda_p(A):=\big\{x\in\mathbb{C}^{\mathbb{N}}\colon\,\,\|(x_ja_n(j))_{j\in\mathbb{N}}\|_{\ell_p}<\infty\,\,\text{for all}\,\,n\in\mathbb{N}\big\}\] if $1\leqslant p\leqslant\infty$ and
\[\lambda_0(A):=\big\{x\in\mathbb{C}^{\mathbb{N}}\colon\,\,\lim_{j\to\infty}x_ja_n(j)=0\,\,\text{for all}\,\,n\in\mathbb{N}\big\}\] which, additionally, possess the structure of an algebra. We will provide natural examples of such objects (e.g. Hadamard algebra) and present sketches of proofs of the following results.

Theorem 1. Let $1\leqslant p<\infty$. TFAE:
  (i) $\lambda_p(A)$ is amenable,
  (ii) $\lambda_p(A)$ is contractible,
  (iii) $\lambda_p(A)$ is unital,
  (iv) $\lambda_p(A)$ is nuclear and $A$ is bounded (i.e. $a_n\in\ell_{\infty}$ for all $n\in\mathbb{N}$).

Theorem 2. TFAE:
  (i) $\lambda_0(A)$ is amenable,
  (ii) $\lambda_{\infty}(A)$ is amenable,
  (iii) $\lambda_{\infty}(A)$ is unital,
  (iv) $A$ is bounded.

The proofs require a new approach and the Banach algebra case cannot be automatically adapted. Especially the second result requires specifically Fréchet algebra techniques.

The study of extreme contractions between Banach spaces is a classical area of research in the geometry of Banach spaces. Norm attainment set of a bounded linear operator between Banach spaces plays a very important role in the study of extreme contractions between Banach spaces. We will explore the connection between an extreme contraction and its norm attainment set, when the domain space is a finite-dimensional polygonal Banach space.

In recent times, the study of the norm attainment set of a bounded linear operator between Banach spaces has been a topic of considerable interest and developments. In this talk, I would like to explore the various facets of this problem, including the case of bounded linear operators between Hilbert spaces and Banach spaces. We would show that it is possible to completely characterize Euclidean spaces among Minkowski spaces, in terms of the operator norm attainment set. We would further explore the norm attainment set of a bounded linear operator between Banach spaces. Using the concept of Birkhoff-James orthogonality and semi-inner-products in Banach spaces, we will completely characterize the operator norm attainment set in the setting of Banach spaces. If time permits, we would also like to briefly mention the various areas of application of the norm attainment set of a bounded linear operator, including the study of extreme contractions.

Let $C_{2\pi}$ be the space of all $2\pi$-periodic continuous real functions defined on $\mathbb{R}$, provided with the sup norm $\Vert \cdot \Vert$, and let $C^r_{2\pi}$ be the space of all $2\pi$-periodic functions which have a continuous derivative of order $r$.

In some applications involving modeling of data collected on the surface  of the human brain arises the problem of approximate reconstruction of periodic functions. In many cases the data are taken at  points uniformly distributed on the surface. In particular, the following problem was studied previously.

For each positive integer $N\gt 1$ set $$x_{j,N}=\frac{2\pi j}{N},\qquad j=0,\cdots,N-1$$ and suppose that the values $f(x_{j,N})$ of a function $f\in C_{2\pi}$ are known.

We want  to construct  a trigonometric polynomial $T_n(f)$ of degree $n=n(N)$ such that:

1. $T_n(f)$ is uniquely determined by the values of $f$ at the points $x_{j,N}$ and
2. $\Vert T_{n}(f)-f\Vert\to 0$ as $N\to +\infty$.

A  regularization method  for the polynomial approximation of a function from its approximate values at fixed nodes was proposed by other authors, they gave explicit expressions for the optimal number of nodes in terms of the original error by using some properties of positive linear operators.

It is known that positive linear operators are saturated. If we want to obtain a higher rate of convergence, then others  approximation methods should be used.

For instance, several authors have considered linear combinations of positive linear operators, but their  results can not be used to present numerical algorithms because they are given in terms of  unknown constants. Moreover, for the case of periodic functions the operators studied are of convolution type, while we are looking for discrete operators.

In this talk we  improve the previous results by using a two-terms linear combination of positive linear operators.

We show how to construct, out of a certain basis, a second biorthogonal set with  similar properties.

Finally we apply the procedure  to coherent states and we consider a simple application of our construction to pseudo-hermitian quantum mechanics.

We define the angle of a bounded linear operator $A$, along an unbounded path emanating from the origin and use it to characterize range-kernel  complementarity. In particular we show that if $0$ faces the unbounded  component of the resolvent set, then $X=R(A)\oplus N(A)$ if and only if  $R(A)$ is closed and some angle of $A$ is less than $\pi$.