IWOTA 2019

International Workshop on

Operator Theory and its Applications

IWOTA 2019

International Workshop on

Operator Theory and its Applications

July 22-26, 2019

Instituto Superior Técnico

Lisbon, Portugal

Instituto Superior Técnico

Lisbon, Portugal

- Mohamed Bendaoud

Moulay Ismail University ## Radial unitary similarity functions on operators and preservers problems

Let $\mathfrak{S}$ be a subset of the algebra $\mathcal{L}(\mathcal{H})$ of all bounded linear operators on an infinite-dimensional complex Hilbert space $\mathcal{H}$ containing all rank one operators. In this talk, we determine the structures of nonlinear transformations on $\mathfrak{S}$ that respect the radial unitary similarity functional values of certain algebraic operations on operators. Related problems on the space of matrices are also discussed.

- Priyadarshi Dey

University of Memphis ## Bicircular projections on some Operator spaces

A projection $P$ on a Banach space is said to be bicircular if $e^{itP}$ is a surjective isometry, for every $t \in \mathbb{R}$. In this talk, I shall present a characterization of these projections on $B(\mathcal{H}, \mathcal{K})$ for $\mathcal{H}$ and $\mathcal{K}$ two Hilbert spaces. Moreover, our characterization also extends to bicircular projections on $B(X,Y)$, for a large class of pairs of Banach spaces.

This talk is based on some joint work with Fernanda Botelho & Dijana Ilišević.

- Yuta Enami

Niigata University ## Maps preserving range of difference of continuous functions with values in a locally convex space

Let $X,T$ be compact Hausdorff spaces, and let $E$ be a locally convex space. We characterize maps $T:C(X,E)\to C(Y,E)$ which stisfies $\mathrm{Ran}(TF-TG)\subset\mathrm{Ran}(F-G)$ for every $F,G\in C(X,E)$. These maps are automatically linear and represented as composition operators.

- Gyorgy Geher

University of Reading, UK ## Isometric maps on spaces of measures III. - Wasserstein spaces over the real line

In this talk I will show a characterisation of isometric embeddings of the $p$-Wasserstein space on the real line. We will start with tha case when $p=1$, and provide the characterisation under the assumption that the map is bijective. We will illustrate with some examples what can go wrong in the non-bijective case. Then we continue with the case when $1\lt p \lt 2$ or $2\lt p$, in which case we show a characterisation also for non-bijective distance preserving maps. Finally, we discuss the $p=2$ case, which is special in some sense. A description of the isometry group was provided in 2010 by Kloeckner, however, an exact formula for the action of the isometries were not known. Here we show an explicit formula for the action of a general isometry in the $p=2$ case.

- Ilja Gogic

University of Zagreb ## The cb-norm approximation of derivations and automorphisms by elementary operators

An attractive and fairly large class of completely bounded (cb) linear maps on $C^*$-algebras that preserve their ideals is the class of elementary operators, that is, those that can be expressed as a finite sum of two-sided multiplications $x \mapsto axb$. Motivated by the fact that derivations and automorphisms of $C^*$-algebras are also completely bounded, we consider which derivations and automorphisms of $C^*$-algebras admit the cb-norm approximation by elementary operators.

- Maliheh Hosseini

Faculty of Mathematics, K. N. Toosi University of Technology ## Reflexivity of The Isometry Group of Bounded Variation Function Spaces

For an arbitrary subset $X$ of the real line with at least two points, let $BV(X)$ be the Banach space of all functions of bounded variation on $X$ endowed with the norm $\|\cdot\|_\infty+ \mathcal{V}(\cdot)$, where $\|\cdot\|_\infty$ and $\mathcal{V}(\cdot)$ denote the supremum norm and the total variation of a function, respectively. Our aim is to show that the group of all surjective linear isometries of $BV(X)$ is topologically reflexive.

- Dijana Ilisevic

University of Zagreb, Croatia ## Isometries, projections and related maps on JB*-triples

A JB*-triple is one of those remarkable structures in which surjective linear isometries are related to corresponding algebraic isomorphisms. Some important JB*-triples are: (matrix and) operator spaces $B(\mathcal{H}, \mathcal{K})$ of bounded linear operators from a complex Hilbert space $\mathcal{H}$ to a complex Hilbert space $\mathcal{K}$, $A(\mathcal{H})$ of skew-symmetric operators on $\mathcal{H}$ and $S(\mathcal{H})$ of symmetric operators on $\mathcal{H}$, $C^\ast$-algebras, etc. The aim of this talk is to recall and connect some recent and not so recent results arising from the fact that a bijective linear operator between JB*-triples is an isometry if and only if it preserves the Jordan triple product, and also to address some open questions.

Some parts of this talk are based on joint work with several collaborators. The work of Dijana Ilisevic has been fully supported by the Croatian Science Foundation under the project IP-2016-06-1046.

- Hironao Koshimizu

National Institute of Technology, Yonago College ## Surjective isometries on spaces of $n$-times continuously differentiable functions

Let $C^{(n)}[0, 1]$ be the linear space of $n$-times continuously differentiable functions on the closed unit interval $[0, 1]$. We consider several norms on $C^{(n)}[0, 1]$, and we give the characterization of surjective isometries on $C^{(n)}[0, 1]$.

- Aziz Lahssaini

Sidi Mohamed Ben Abdellah ## On the pseudospectrum preservers

Let $\mathcal{B}(X)$ be the algebra of all bounded linear operators on a complex Banach space $X$. We characterize additive maps from $\mathcal{B}(X)$ onto $\mathcal{B}(Y)$ compressing the pseudospectrum subsets $\Delta_{\epsilon}(.)$, where $\Delta_{\epsilon}(.)$ stands for any one of the spectral functions $\sigma_{\epsilon}$, $\sigma_{\epsilon}^l$ and $\sigma_{\epsilon}^r$ for some $\epsilon \gt 0$.

- Yingfen Lin

Queen ## Isomorphisms of $C^\ast$-algebras on Lie groups

I will briefly introduce the group $C^\ast$-algebra and give some examples on abelian groups and a class of exponential Lie groups. Afterward I will talk about the isomorphism problem for $C^\ast$-algebras on Lie groups, mainly, to answer when two group $C^\ast$-algebras are isomorphic and to what extent a Lie group is determined by its $C^\ast$-algebra.

- Takeshi Miura

Niigata University, Japan ## Surjective isometries on a Lipschitz space of analytic functions

Let $H(\mathbb D)$ be the linear space of all analytic functions on the open unit disc $\mathbb D$. We define $\mathcal S^\infty$ by the linear subspace of all $f \in H(\mathbb D)$ with bounded derivative $f'$ on $\mathbb D$. We give the characterization of surjective, not necessarily linear, isometries on $\mathcal S^\infty$ with respect to the following two norms: $\| f \|_\infty + \| f' \|_\infty$ and $|f(a)| + \| f' \|_\infty$ for $a \in \mathbb D$, where $\| \cdot \|_\infty$ is the supremum norm on $\mathbb D$.

- Lajos Molnar

University of Szeged, and Budapest University of Technology and Economics ## Maps preserving means on positive definite or positive semidefinite cones in operator algebras

Means of positive definite and positive semidefinite matrices or operators can be considered as operations. Therefore, transformations respecting them are kinds of morphisms. In this talk we survey former results describing the structures of such maps and also present recent ones in different settings, stretching from the case of matrix algebras to abstract $C^*$-algebras.

- Monika

University of Memphis ## Quotient operators on tensor product spaces

Let $X$, $Y$, $Z$ are Banach spaces. An operator $Q : Z\rightarrow Y$ is a quotient operator if $Q$ is surjective and $\|y\|= \inf\{\|z\|\ : z\in Z, Q(z)=y \}$ for every $y \in Y$. Also let $I$ be an identity operator on $X$. In this talk I will mention some conditions under which the maps $I \otimes Q$ and $Q \otimes I$ are again quotient operators on the respective tensor product spaces. Also some related results and recent developments will be presented. This is a joint work with TSSRK Rao.

- Michiya Mori

University of Tokyo ## Order isomorphisms of von Neumann algebras

In this talk, I will give a complete description of order isomorphisms between intervals of von Neumann algebras. For this description, Jordan *-isomorphisms and locally measurable operators play crucial role. In particular, I will explain that every order isomorphism between self-adjoint parts of two von Neumann algebras without commutative direct summands is affine. This description generalizes several previous works on type I factors by L. Molnar and P. Semrl.

- Mustapha Sarih

University of Meknes ## Maps leaving the pseudo spectra of operators invariant

Let $\mathcal{H}$ be a complex Hilbert space, and let $\mathcal{L}(\mathcal{H})$ be the space of all bounded linear operators on $\mathcal{H}$.

In this talk we present new results concerning nonlinear maps on $\mathcal{L}(\mathcal{H})$ leaving invariant the pseudo spectrum of operators. The corresponding results for the pseudo spectral radius of operators are also presented.

- Zsigmond Tarcsay

Eotvos Lorand University, Department of Applied Analysis and Computational Sciences ## Maps preserving absolute continuity and singularity of positive operators

Let $H$ be a complex Hilbert space and denote by $B(H)_+$ the set of all positive operators on $H$. We say that $A \in B(H)_+$ is absolutely continuous with respect to $B\in B(H)_+$ if, for every sequence $(x_n)$ in $H$, $(A(x_n−x_m), x_n−x_m)\to 0$ and $(Bx_n, x_n) \to 0$ imply $(Ax_n, x_n) \to 0$. On the other hand we say that $A,B$ are mutually singular if $C=0$ is the only positive operator such that $C\leq A$ and $C\leq B$. The aim of this talk is to describe the general form of those bijective maps $\phi : B(H)_+ \to B(H)_+$ which preserve absolute continuity, respectively, singularity in both directions.

- Tamás Titkos

Alfred Renyi Institute of Mathematics ## Isometric maps on spaces of measures I. - Banach-Stone type theorems

In this talk I will survey some recent results on surjective isometries of various different metric spaces of probability measures. I call them Banach-Stone type theorems, because all these theorems are saying that the isometry group can be described by morphisms of the underling structure. The range of examples includes the Lévy-, Kolmogorov-Smirnov-, Kuiper-, and Lévy-Prokhorov metrics. Closing the talk I will demonstrate that things could become very complicated when one drops the surjectivity condition and tries to characterize isometric embeddings of Wasserstein spaces with discrete underlying space.

The talk is based on a joint work with György Pál Gehér (University of Reading) and Dániel Virosztek (IST Austria).

- Daniel Virosztek

Institute of Science and Technology Austria ## Isometric maps on spaces of measures II. - Wasserstein spaces over the interval

I will report on some aspects of our study of Wasserstein isometries --- a joint work with György Pál Gehér (University of Reading) and Tamás Titkos (Rényi Institute, Budapest).

More precisely, I will present the description of the isometry group of the Wasserstein space over the interval for all parameters $p \geq 1.$

We will see that the exceptional parameter value is $p=1$; the isometry group of $\mathcal{W}_1([0,1])$ is the Klein group $C_2 \times C_2,$ while we have isometric rigidity for p>1.

Some isometries of $\mathcal{W}_1([0,1])$ even split mass. We never use the bijectivity assumption in our arguments, so --- as a byproduct --- we obtain that the (a priori non-surjective) isometric self-embeddings of Wasserstein spaces over the interval are necessarily bijective isometries.