**Events of Functional Analysts**

** at NSYSU, 2016**

Department of
Applied Mathematics

National Sun Yat-sen University,
Kaohsiung, Taiwan.

台灣•高雄•中山大學•應用數學系

Several functional analysts are visiting National Sun Yat-sen
University and offering workshops below.
All interested are welcome to join.

I.
Mini-workshop
in Functional Analysis, Day 1,

**Thursday,
May 19, 2016. Venue: SC-4013, NSYSU**

**1.
**L.
Molnár (University of Szeged, Hungary):
Transformations on positive matrices and operators (13:10-13:55pm)** **

We survey results concerning different sorts of
transformations on the cones of positive semidefinite
or positive definite matrices and operators. The transformations in question
are maps preserving certain important operations, or distances (or even
so-called generalized distance measures), or partial orders. Hence, they are
kinds of isomorphisms, or "generalized" isometries. In the talk we shall describe their precise
structures.

**2.
**D.
Virosztek (Budapest University of Technology and
Economics, Hungary): Continuous Jordan triple endomorphisms
of _{} (14:00-14:45pm)

We describe the structure of all continuous Jordan
triple endomorphisms of the set _{} of all positive definite 2x2 matrices. We
also mention an application concerning sorts of surjective
generalized isometries on _{} and, as second application, we
complete a former result on the structure of sequential endomorphisms
of finite dimensional effect algebras. Furthermore, we characterize the
continuous endomorphisms of the space of
three-dimensional velocity vectors with respect to the Einstein addition (or
relativistic sum). Joint work with Lajos Molnár.

**3.
**Gy. P. Gehér (University of Szeged, Hungary):
Is it possible to determine a point lying in a simplex if we know the distances
from the vertices? (15:00-15:45pm)

It is an elementary fact that if we fix an arbitrary
set of _{} affine
independent points _{} in _{}, then the Euclidean distances _{} determine the
point _{} in _{} (and therefore in
the simplex _{}) uniquely. In my
talk I would like to investigate a similar problem in general normed spaces. Npmely, I will present a characterization of those, at
least _{}-dimensional, real normed spaces _{} for which every
set of _{} affine
independent points _{}, the distances _{} determine the
point _{} lying in the
simplex _{} uniquely.
Surprisingly, the characterization depends on _{}.

**4.
**Hongke Du 杜鴻科 (Shannxi Normal
University, China): Uniteries of Krein
space _{} (15:55-16:40pm)

In this talk, we will discuss the block-operator
expression and connectivity of Uniteries in Krein space .

**5.
**Denny Leung 梁浩瀚(National University of Singapore): Inverses of disjointness
preserving operators (16:50-17:35pm)** **

A linear operator between (possibly vector-valued)
function spaces is disjointness preserving if it maps
disjoint functions to disjoint functions.
Here, two functions are said to be disjoint if at least one of them
vanishes at each point. Linear disjointness preserving operators have been well
studied. A particular question of
interest is when the inverse of a disjointness
preserving linear isomorphism must also be disjointness
preserving. In this talk, we
consider this question for operators on various types of function spaces,
including spaces of (little) Lipschitz functions,
uniformly continuous functions and differentiable functions. It is shown that a disjointness
preserving linear isomorphism whose domain is one of these types of spaces
(scalar-valued) has a disjointness preserving
inverse, subject to some topological conditions on the range space. A representation for a general linear disjointness preserving operator on a space of
vector-valued functions is also given.

This talk is based on joint work with Li Lei (Nankai U) and Wang Ya-shu
(National Chung Hsing U).

II.
Mini-workshop
in Functional Analysis, Day 2,

**Tuesday,
May 24, 2016.** **Venue: SC-4011, NSYSU**

**6.
**Hao-Wei Huang (NSYSU, Taiwan): Analytic and combinatorial aspects of bi-freely
infinitely divisible laws (14:10-14:55pm)** **

In free probability the notion of free convolution of
probability distributions on has played an important role since its
inception by D. Voiculescu some 30 years ago. In
2013, Voiculescu generalized the notion of free
independence to study left and right actions on reduced free product spaces
simultaneously, known as bi-free independence. One generalization of the free
convolution to the bi-free setting is the bi-free convolution of planar
probability distributions. In this talk, we will explain that the bi-freely
infinitely divisible laws, and only these laws, can be used to approximate the
distributions of sums of identically distributed bi-free pairs of commuting
faces. We will also talk about bi-free Levy-Khintchine
representations from an infinitesimal point of view. The proofs depend on the
bi-free harmonic analysis machinery that we developed for integral transforms
of two variables, and the combinatorics of moments
and bi-free cumulants. If time permits, some recent
developments in this direction will also be discussed.

**7.
**D.
Virosztek (Budapest University of Technology and
Economics, Hungary): Maps on quantum states and positive definite matrices
preserving Bregman and Jensen divergences
(15:00-15:45pm)

We describe the structure of all bijective
maps on the state space of a finite quantum system which preserve Bregman or Jensen divergences. Similar results concerning
the cone of positive definite matrices are also given. We would like to
highlight how essentially different techniques lead to the description of the
generalized isometries in the two aforementioned
cases. Partially based on a joint work with Lajos Molnár and József Pitrik.

**8.
**Gy. P. Gehér (University of Szeged, Hungary): Isometries of Grassmann spaces
(16:00-16:45pm)

Botelho, Jamison, and Molnár
have recently described the general form
of surjective isometries of Grassmann spaces on
complex Hilbert spaces under certain dimensionality assumptions. In this talk
we provide a new approach to this problem which enables us first, to give a
shorter proof and second, to remove dimensionality constraints completely. In
one of the low dimensional cases, which was not covered by Botelho,
Jpmison, and Molnár,
an exceptional possibility occurs. As a byproduct, we are able to handle the
real case as well. Furthermore, in finite dimensions we remove the surjectivity assumption. A variety of tools is used in
order to achieve our goal.

**9.
**Hongke Du杜鴻科(Shannxi Normal
University, China): Block-operator expression and connectivity of Lagrangian Grassmannian _{} (16:50-17:35pm)

In this talk, we will use the block-operator technique
and spectral theory of operators to study
the expression and connectivity of Lagrangian Grassmannian .

**Sponsors:**
National Center of Theoretical Sciences (Math, Taipei), Ministry of Science and
Technology, and National Sun Yat-sen University.

Contact:

Ngai-Ching Wong黃毅青, wong@math.nsysu.edu.tw,
Tel: (886) 7-525-2000 ext. 3818,

Chia-Feng Yen 嚴嘉鳳, yencf@math.nsysu.edu.tw, Tel: (886)
7-5252000 ext. 3849; or visit