General
Registration for Positivity VII is now open. The deadline for registration, submission of abstracts and payment of the conference fee of 200 EUR is Friday June 7th, 2013. The conference fee includes coffee/tea, lunch, as well as a combined excursion and conference dinner. After June 7th registration is still possible at an increased fee of 250 EUR, but without the possibility of giving a contributed talk.
The seventh Positivity conference will be held from July 22-26, 2013, at the science campus of Leiden University, The Netherlands, jointly organized by Leiden University and Delft University of Technology. It is the Zaanen Centennial Conference, on the occasion of the 100th birth year of Adriaan Cornelis Zaanen, who held the chair of functional analysis in Leiden for more than 25 years, and played a prominent role in Positivity during that period.
The organizing committee consists of Marcel de Jeu (Leiden, chair), Ben de Pagter (Delft), Miek Messerschmidt (Leiden), Jan Rozendaal (Delft), Onno van Gaans (Leiden), and Mark Veraar (Delft).
Invited speakers, all confirmed:
- Francesco Altomare (Bari, Italy)
- Wolfgang Arendt (Ulm, Germany)
- Karim Boulabiar (Tunis, Tunisia)
- Qingying Bu (Mississippi, USA)
- Guillermo Curbera (Sevilla, Spain)
- Julio Flores (Madrid, Spain)
- Yehoram Gordon (Haifa, Israel)
- Rien Kaashoek (Amsterdam, The Netherlands)
- Coenraad Labuschagne (Johannesburg, South Africa)
- Boris Mordukhovich (Detroit, Michigan, USA)
- Ioannis Polyrakis (Athens, Greece)
- Abdelaziz Rhandi (Salerno, Italy)
- Evgeny Semenov (Voronezh, Russia)
- Fedor Sukochev (Sydney, Australia)
- Jun Tomiyama (Tokyo, Japan)
- Jan van Neerven (Delft, The Netherlands)
All participants will be given the opportunity for a contributed talk of 30 minutes.
Leiden is 15 minutes by train from Amsterdam Airport (Schiphol), with trains running every quarter of an hour.
For inquiries, or inclusion in the mailing list for this conference, please write to positivity2013@gmail.com.
Starting the week following Positivity VII, the biyearly Banach algebra conference will be held in relatively nearby Gothenburg, Sweden.
Announcements
May 13th, 2013. Positivity VII conference, Third and final general announcement
Dear all,
This is the third and final general announcement for Positivity VII, a conference devoted to ordered structures and their applications. From now on further announcements will only be sent to participants who have registered on the website. The conference will be held July 22-26, 2013 at the science campus of Leiden University, the Netherlands. It is jointly organized by Leiden University and Delft University of Technology, and is also the "Zaanen Centennial Conference," celebrating 100 years since the birth of Adriaan Cornelis Zaanen. We are happy to announce that over 110 participants have already registered.
REGISTRATION
You can register on the website http://websites.math.leidenuniv.nl/positivity2013/ Registration is required for all participants, including pre-registered participants. Deadline for early registration is June 7, 2013, at a conference fee of 200 euros and with the possibility of giving a talk. Registration after June 7th is possible at a fee of 250 euros but without the possibility of giving a talk. Registration will only be completed once payment of the conference fee has been received. Payment details can be found on the website.
CONTRIBUTED TALKS
Contributed talks are 30 minutes. Blackboards and beamer facilities will be available, overhead projectors will not. Abstracts can be submitted in plain latex on http://websites.math.leidenuniv.nl/positivity2013/. Deadline for submission of abstracts is June 7, 2013.
HOTEL RESERVATION
If you have not yet made a hotel reservation, we strongly recommend you to do so as soon as possible. During the summer months hotels can be fully booked. A selection of suitable hotels can be found on the website http://websites.math.leidenuniv.nl/positivity2013/.
VISA
If you require an invitation for a visa, please do not hesitate to contact us at positivity2013@gmail.com. We ask participants who have not yet received a visa to register for the conference nevertheless. Registration will not yield any consequences if no visa can be obtained.
We look forward to your registration and participation in Positivity VII.
Kind regards,
the organizing committee,
Marcel de Jeu (Leiden, chair)
Ben de Pagter (Delft)
Miek Messerschmidt (Leiden)
Jan Rozendaal (Delft)
Onno van Gaans (Leiden)
Mark Veraar (Delft).
Mar 4th, 2013. Positivity VII conference, Second announcement
Dear all,
This is the second announcement for Positivity VII, a conference devoted to ordered structures and their applications. The conference will be held July 22-26, 2013 at the science campus of Leiden University, the Netherlands. It is jointly organized by Leiden University and Delft University of Technology, and is also the "Zaanen Centennial Conference," celebrating 100 years since the birth of Adriaan Cornelis Zaanen. We are happy to announce that over 130 participants have already pre-registered.
REGISTRATION
Registration is open starting today. You can register on the website http://websites.math.leidenuniv.nl/positivity2013/. Registration is required for all participants, including pre-registered participants. Deadline for early registration is June 7, 2013, at a conference fee of 200 euros and with the possibility of giving a talk. Registration after June 7th is possible at a fee of 250 euros but without the possibility of giving a talk. Registration will only be completed once payment of the conference fee has been received. Payment details can be found on the website.
CONTRIBUTED TALKS
Contributed talks are 30 minutes. Blackboards and beamer facilities will be available, overhead projectors will not. Abstracts can be submitted in plain latex on http://websites.math.leidenuniv.nl/positivity2013/. Deadline for submission of abstracts is June 7, 2013.
HOTEL RESERVATION
If you have not yet made a hotel reservation, we strongly recommend you to do so as soon as possible. During the summer months hotels can be fully booked. A selection of suitable hotels can be found on the website http://websites.math.leidenuniv.nl/positivity2013/.
VISA
If you require an invitation for a visa, please do not hesitate to contact us at positivity2013@gmail.com.
We look forward to your registration and participation in Positivity VII.
Kind regards, the organizing committee,
Marcel de Jeu (Leiden, chair) Ben de Pagter (Delft) Miek Messerschmidt (Leiden) Jan Rozendaal (Delft) Onno van Gaans (Leiden) Mark Veraar (Delft).
Dec 4th, 2012. Positivity VII conference, first announcement
Dear all,
This is the first announcement for Positivity VII, a conference devoted to ordered structures and their applications. The conference will be held on July 22-26, 2013 at the science campus of Leiden University, the Netherlands. It is jointly organized by Leiden University and Delft University of Technology and is also the "Zaanen Centennial Conference", the conference celebrating 100 years since the birth of Adriaan Cornelis Zaanen.
PRE-REGISTRATION
Pre-registration is open starting today. You can pre-register on the website http://websites.math.leidenuniv.nl/positivity2013/ . Pre-registration carries no obligations but is appreciated. If you require an invitation for a visa or financial support, please feel free to contact us at positivity.2013@gmail.com.
HOTEL RESERVATION
We strongly recommend making hotel reservations in Leiden as soon as possible. During the summer months hotels are usually fully booked. A selection of suitable hotels can be found on the website http://websites.math.leidenuniv.nl/positivity2013/.
We are looking forward to your pre-registration and your participation in Positivity VII.
Kind regards,
the organizing committee,
Marcel de Jeu (Leiden, chair) Ben de Pagter (Delft) Miek Messerschmidt (Leiden) Jan Rozendaal (Delft) Onno van Gaans (Leiden) Mark Veraar (Delft).
Registration
Submit Title and Abstract
Conference Fee
Registration will only be completed once payment of the conference fee has been received. The deadline for payment of the conference fee of 200 EUR is Friday June 7th, 2013. Thereafter the conference fee is 250 EUR. Please take possible transaction fees into account when making your payment.
Account number: 542891514
Account holder: TU Delft, SSC F&C, faculteit EWI
Postal address of account holder: Mekelweg 4, 2628CD, Delft, The Netherlands
IBAN: NL11ABNA0542891514
SWIFT/BIC: ABNANL2A
Bank: ABN AMRO
Postal address of bank: Gustav Mahlerlaan 10, 1082PP, Amsterdam, The Netherlands
Transaction Description: BAAN:IWD07G:POSITIVITY2013:[Surname+GivenName]
Transaction DiExample: BAAN:IWD07G:POSITIVITY2013:DoeJohn
If you bring non-participating guests who would like to have meals and/or participate in the excursion and conference dinner, please contact the organizers at positivity2013@gmail.com as this will involve extra costs.
Hotels
Hotels in Leiden may be fully booked or become very expensive when the summer is approaching, so we advise you to make reservations at your earliest convenience.
Location (see map below)
The conference venue is on the Science Campus of Leiden University, which is approximately 1,5 km from the main railway station Leiden Centraal. On the campus there is not much more than university buildings. For restaurants, shops, pubs and more you have to go to downtown Leiden. Downtown Leiden starts at the other side of the railway station and has a diameter of roughly 2 km. The quickest way of getting around is by bike (see bike rental options below) or on foot. There are bus services in town and to the university, but taking waiting time into account they are often not much faster than walking. Some hotels are somewhat closer to the conference venue than others but considerably further from downtown Leiden. Make sure you choose a hotel that fits your priorities.
List of some hotels
Below is a list of some hotels that you may want to consider. Some hotels offer special rates for the positivity conference or guests of Leiden University, and these rates are then mentioned below. The availability of these rates can be limited and they may be changed as time progresses. Please contact the hotel directly for up-to-date information and for their cancellation policy.
Please note that your own favourite booking site (such as booking.com) could in some cases offer better rates, even better than the special rates below, and is worth a check.
All prices below are in euro including VAT, but excluding an additional tourist tax of 2,00 euro per person per night.
Hotel Ibis Leiden Centre
Stationsplein 240-242
2312 AR LEIDEN
E-mail: H8087-RE@accor.com
Special rate 1p or 2p room excluding breakfast 84,00 per night (same rate for 1p as for 2p), breakfast 15,00 per person per night.
For reservations at this special rate do not use a website, not even the hotel's, but contact the hotel directly by telephone or email and refer to the Positivity conference at Leiden University and Onno van Gaans as one of the organizers.
Ibis is in a new building on the square in front of the railway station.
Tulip Inn Leiden Centre/Golden Tulip Leiden Centre
Schipholweg 3
2316 XB Leiden
Telephone number: +31 71 4083500
Faxnumber: +31 71 5226675
E-mail: reservations@goldentulipleidencentre.nl
Tulip Inn: special rate 1p or 2p standard room including breakfast buffet 79,00 per night (same rate for 1p as for 2p, including breakfast)
Golden Tulip: special rate 1p or 2p standard room including breakfast buffet 99,00 per night (same rate for 1p as for 2p, including breakfast)
For reservations at this special rate do not use a website, not even the hotel's, but contact the hotel directly by telephone or email and refer to the Positivity conference at Leiden University and Onno van Gaans as one of the organizers.
Tulip Inn and Golden Tulip are a three star and a four star hotel in adjacent buildings, very close to the railway station.
Holiday Inn Leiden
Haagse Schouwweg 10
2300 PA Leiden
Telephone +31 71 5355555
Fax +31 71 5355553
Special rate 1p room including breakfast 97,50 per night.
For reservations at this special rate do not use a website, not even the hotel's, but fill out the reservation form and send it by fax to the hotel.
Holiday Inn is closest to the conference venue. The hotel offers bike rental.
City Hotel Nieuw Minerva
Boommarkt 23
2311 EA Leiden
Telephone +31 71 512 6358
E-mail: hotel@nieuwminerva.nl
Special rate 1p room with shower and WC including breakfast 79,00 euro per night
Special rate 2p room with shower and WC including breakfast 99,00 euro per night
For reservations at this special rate do not use a website, not even the hotel's, but contact the hotel directly by telephone or email and refer to the Positivity conference at Leiden University and Onno van Gaans as one of the organizers.
Nieuw Minerva is in the city centre, located in six 16th-century canal houses.
Best Western City Hotel Leiden
Lange Mare 43
2312 GP Leiden
Tel : +31 71- 5130505 Fax : +31 71- 5131626
Best Western is a three star hotel with 39 rooms in the city centre.
Leiden Hilton Garden Inn
Willem Einthovenstraat 3
2342 BH Oegstgeest
Tel +31 71 7111000
Hilton Garden Inn is quite a new building next to the university campus, with the highway A44 in between.
Van der Valk Hotel Leiden
Haagse Schouwweg 14
2332 KG Leiden
Tel +31 (0)71 5 731 731
Fax +31 (0)71 5 731 710
E-mail: leiden@valk.nl
Special rate 1p room with bath, separate shower and WC including breakfast 97,00 euro per night
Special rate 2p room with bath, separate shower and WC including breakfast 109,50 euro per night
Van der Valk is a four star hotel with 80 rooms & suites next to the Old Rhine and is situated nearby the highway A44. The hotel offers bike rental.
Please use the following reservation form.
Bastion hotel Oegstgeest
Rijnzichtweg 97
2342 AX Oegstgeest
Tel.: +31 (0)71-5153841
Fax.: +31 (0)71-5154981
Bastion hotel Oegstgeest is situated along the highway A44 in the village Oegstgeest
Smaller hotels
There are also several smaller hotels in Leiden: Rembrandt Hotel Leiden, Marienpoel Hotel Leiden, Hotel Mayflower, De Doelen, Huys van Leiden.
Map and distances
Here are a map with the locations of the hotels and a table with the distance and walking time to the conference venue and the railway station.
View on Google maps Printable version
| Abbrev. | Hotel | distance to conference | distance to railway station |
| Ib | Hotel Ibis | 1,5 km (18 min) | 0,3 km (3 min) |
| Tu | Tulip Inn/Golden Tulip | 1,7 km (21 min) | 0,3 km (3 min) |
| Ho | Holiday Inn | 1,0 km (11 min) | 2,4 km (29 min) |
| NM | Nieuw Minerva | 2,2 km (26 min) | 1,0 km (13 min) |
| BW | Best Western City Leiden | 2,3 km (27 min) | 0,8 km (10 min) |
| Ga | Hilton Garden Inn | 1,3 km (16 min) | 2,4 km (29 min) |
| Va | Van der Valk | 1,5 km (18 min) | 3,0 km (36 min) |
| Ba | Bastion Oegstgeest | 2,2 km (26 min) | 2,5 km (30 min) |
Times are in minutes at average walking speed.
Taxis
There is a taxi stand at the city centre side of the railway station. Most taxi companies charge 2,95 initial fee plus 1,95 per km plus 0,32 per minute. For a ride from the station to downtown or to the university campus expect prices in the range of 8 to 15 euro.
Bike rental
There are bikes available for rental at the bicycle shop/parking at the campus side of the railway station building: Fietspoint Oldenburger. Prices are about 7,50 euro per bike per day or 37,50 per week. Rentals have to be paid in advance with a cash deposit of 50 euro and valid identification has to be shown. Availability is limited. Reservations can only be made by telephone: +31 (0)71-512 0068. Please inquire at your hotel whether secure bike parking is available.
Visa
Depending on your nationality, you may require a visa to enter the Netherlands. Information on whether you require a visa, and how to apply for one, can be found on this website.Registered Participants
Plenary Lectures
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On positive linear operators preserving polynomials
Francesco Altomare
Bari, Italy
The talk will be centered about a special class of positive linear operators acting on the space $C(K)$ of all continuous functions defined on a convex compact subset $K$ of $\mathbf{R}^{d}$, $d\geq 1$, having nonempty interior. Actually, this class consists of all positive linear operators $T$ on $C(K)$ which preserve polynomials on $K$ and which, in addition, leave invariant the continuous affine functions on $K.$
The interest for such operators comes from the study of the differential operator $W_{T}$ naturally associated with $T$ which is defined as
\[ W_{T}(u):=\sum\limits_{i,j=1}^{d}\alpha _{ij}\frac{\partial ^{2}u}{\partial x_{i}\partial x_{j}} \]$(u\in C^{2}(K))$, where $\alpha _{ij}(x):=T(pr_{i}pr_{j})(x)-(pr_{i}pr_{j})(x)$ ( $i,j=1,\ldots ,d$ and $% x=(x_{i})_{1\leq i\leq d}\in K).$
The differential operator $W_{T}$ is elliptic and it degenerates on the set of the extreme points $\partial _{e}K$ of $K$. Because of the special assumptions on $T,$ it turns out that $(W_{T},C^{2}(K))$ is closable in $C(K)$ and its closure generates a Markov semigroups on $C(K)$ which can be represented as a limit of suitable iterates of Bernstein-Schnabl operators associated with $T$.
The main aim of the talk is to discuss the existence of such operators in the special case when $K$ is strictly convex, i.e., $\partial _{e}K$ $=$ $% \partial K.$ In this same setting we also give a complete characterization of positive projections on $C(K)$ which preserve polynomials.
For more details and for several other aspects related to this theory the reader is referred to the forthcoming monograph [1].
[1] F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Ra\c{s}, Differential Operators, Markov Semigroups and Positive Approximation Processes Associated with Markov Operators, in preparation.
Positive solutions of evolution equations governed by forms
Wolfgang Arendt
Ulm, Germany
Form methods play an important role to solve evolution equations. We will give an introduction starting by the generation theorem based on the Lax-Milgram lemma. In applications the underlying Hilbert space is an $L^2$ space and we may ask when solutions with positive initial value remain positive for all time. In the case of symmetric forms the famous Beurling-Deny criterion gives a characterization.
In this talk we will show how positivity of the solutions of evolution equations governed by (not necessarily symmetric) non-autonomous forms can be characterized. These recent results obtained in collaboration with D. Dier and E.M. Ouhabaz generalize the classical Beurling-Deny criterion to situations which are most common in applications. More generally we will characterize when arbitrary closed convex sets are invariant. A particular case is the submarkovian property. Even in the finite-dimensional case new interesting results for ode can be deduced from our criterion.
Finally we will discuss local forms and semigroups of local operators. Here a beautiful result of Professor Zaanen will play an important role.
References.
W. Arendt, S. Thomaschewski: Local operators and forms. Positivity 9 (2005), 357-367
W. Arendt, D. Dier, E.M. Ouhabaz: Invariance of convex sets for non-autonomous evolution equations governed by forms. Preprint 2013.
Algebraic Order Bounded Disjointness Preserving Operators
Karim Boulabiar
Tunis, Tunisia
A (linear) operator $T$ on a real vector space is said to be algebraic if $\Pi\left( T\right) =0$ for some non-zero real polynomial $\Pi$. This talk discusses algebraic order bounded disjointness preserving operators on an Archimedean Riesz space $E$. One of the major results asserts that, if $T$ is an order bounded disjointness preserving operator on $E$ such that $T\left( E\right) $ is Riesz subspace of $E$, then $T$ is algebraic if and only if there exist integers $0\leq m\leq n$ such that $T^{n!}$ is an $I$-step function on $T^{m}\left( E\right) $. The talk ends up with a few open problems. This research is based upon a joint work with Gerard Buskes and Gleb Sirotkin.
Multilinear Operators and Homogeneous Polynomials on Banach Lattices
Qingying Bu
Oxford, Mississippi, USA
In this talk, we will discuss multilinear operators and homogeneous polynomials on Banach lattices by employing Fremlin positive projective tensor product of Banach lattices, and will study them by using the operator norm, the regular norm, and the norm of bounded variation of multilinear operators and homogeneous polynomials on Banach lattices. As a results, we will obtain when all continuous multilinear operators and homogeneous polynomials on Baanch lattices are regular. We will also provide new AM-spaces and AL-spaces of multilinear operators and homogeneous polynomials.
The Cesàro operator acting on $\ell^p$,
and consequences for Hardy spaces on the disc
Guillermo Curbera
Sevilla, Spain
The Cesàro operator on sequences, given by
\[ a=(a_n)_{0}^\infty\in{\mathbb C}^{{\mathbb N}} \longmapsto \mathcal{\,C}(a):=\Big(\frac{1}{n+1} \sum_{k=0}^n a_k\Big)_{n=0}^\infty\in{\mathbb C}^{{\mathbb N}} , \]is bounded on $\ell^p$, for $1 < p < \infty$. From this starting point several sequence spaces arise; namely:
\[ [\,\mathcal{C},\ell^p]:=\Big\{a=(a_n)_0^\infty\in{\mathbb C}^{{\mathbb N}} : \mathcal{\,C}(a)=\Big(\frac{1}{n+1} \sum_{k=0}^n a_k\Big)_{n=0}^\infty\in\ell^p\Big\} , \]and
\[ ces_p:=\Big\{a=(a_n)_0^\infty\in{\mathbb C}^{{\mathbb N}} : \mathcal{\,C}(|a|)=\Big(\frac{1}{n+1} \sum_{k=0}^n |a_k|\Big)_{n=0}^\infty\in\ell^p\Big\} . \]The discussion of the action of the Cesàro operator on these spaces allows to deduce consequences for the Cesàro operator
\[ f(z)=\sum_{n=0}^\infty a_kz^k\longmapsto \mathcal{C}(f)(z):=\sum_{n=0}^\infty \Big(\frac{1}{n+1}\sum_{k=0}^na_k\Big)z^n \]when acting on the Hardy spaces on the disc, ${H^p({\mathbb D})}$, for $1\le p< \infty$. In this way, it arises the Banach space of analytic functions $[{\cal C},H^p]$ consisting of all analytic functions that ${\cal C}$ maps into $H^p(D)$. It is noteworthy that its elements are characterized (for $ 1 < p < \infty $) by a growth condition:
\[ f\in[{\cal C},H^p] \iff \int_0^{2\pi}\left(\int_0^1\frac{|f(re^{i\theta})|^2} {|1-re^{i\theta}|^2}(1-r)\,dr\right)^{p/2}d\theta < \infty . \]Of particular interest is the subspace $\mathcal{H}(ces_2)$ of $[{\cal C},H^2] $ consisting on those functions which are the unconditional sum of their Taylor series. For this space $\mathcal{H}(ces_2)$ we discuss the multipliers and the spectrum of the Cesàro operator.
The work presented is a collaboration with Werner J. Ricker, from the Katholische Universität Eichstätt--Ingolstadt (Germany).
Disjointly homogeneous spaces: some bits and pieces
Julio Flores
Madrid, Spain
A Banach lattice $E$ is said to be disjontly homogeneous if every pair of normalized disjoint sequences in $E$ have equivalent subsequences. In this talk we consider some aspects around this notion; in particular we are interested in deciding whether being disjointly homogeneous is a selfdual property.
Joint work with F.L Hernandez, E. Spinu, P. Tradacete and V. Troitsky
Applications of the Gaussian Min-Max Theorem
Yehoram Gordon
Haifa, Israel
We show how to apply the Gaussian min-max theorem to provide simple full proofs of several famous results in asymptotic geometric analysis, such as, the Dvoretzky theorem, the Johnson- Lindenstrauss Lemma, Gluskin's theorem on embedding in $\ell^n_1$, the Milman-Schechtman theorem on isomorphic embedding, the restricted isometry property (RIP) for sparse vectors, and more.
State space formulas for rational contractive solutions to a matrix-valued Leech problem
Rien Kaashoek
Amsterdam, The Netherlands
Let $G$ and $K$ be matrix-valued $H^\infty$ functions of sizes $m\times p$ and $m\times q$, respectively. By definition a contractive solution to the Leech problem generated by $G$ and $K$ is a $p\times q$ matrix-valued $H^\infty$ function $X$ satisfying the equation $G(z)X(z)= K(z) \hspace{4pt} (|z|<1)$ and the norm constraint $\sup_{|z|<1}\|X(z)\|\leq 1$. A famous result of R.B. Leech tells us that such a contractive solution exists if and only if $T_GT_G^*-T_KT_K^* \ \mbox{is positive}$, where $T_G $ and $T_K $ are the (block) Toeplitz operators defined by $G$ and $K$, respectively. In this talk we assume additionally that $G$ and $K$ are rational functions. In that case it is known from mathematical system and control theory that $G$ and $K$ admit state space representations of the form:
\[ G(z)= D_1 + z C(I_n - z A)^{-1}B_1, \quad K(z)= D_2 + z C(I_n - z A)^{-1}B_2. \]Here $I_n$ is the $n\times n$ identity matrix, $A$ is a square matrix of order $n$ which has all its eigenvalues in the open unit disc, and $B_1$, $B_2$, $C$, $D_1$ and $D_2$ are matrices of appropriate sizes. Inspired by recent work of T. Trent and S. ter Horst, we shall present a finite dimensional state space procedure to obtain rational contractive solutions to our Leech problem assuming that $G$ and $K$ are given by the above state space representations and the necessary positivity condition is satisfied. Relations with the classical corona problem will be discussed too. The talk is based on joint work with A.E. Frazho and S. ter Horst.
An order-theoretic approach to stochastic processes
Coenraad Labuschagne
Johannesburg, South Africa
It is possible to developed a theory of stochastic processes in Riesz space without using measure theory. In this approach, there is no underlying measure space; instead, the ordering on the Riesz space is used in the development.
Many of the notions and results in the classical setting of stochastic processes on probability spaces have been extended to the Riesz space setting for discrete time stochastic processes. Progress has also been made with this development in the case of continuous time stochastic processes in Riesz spaces.
These ideas are applicable, for example, in Banach spaces, Banach lattices, Bochner spaces and their extensions to the $l$-tensor product of a Banach lattice and a Banach space.
We will give a brief overview of the theory as developed thus far.
Positivity in variational analysis and optimization
Boris Mordukhovich
Detroit, Michigan, USA
We discuss interrelationships between positivity ideas in analysis and novel developments in variational analysis and generalized differentiation of vector-valued and set-valued mappings with values in ordered spaces. Significant progress of these developments has been recently achieved in applications to multiobjective optimization and economic modeling, which will be presented in the talk.
Some topics on the theory of cones
Ioannis Polyrakis
Athens, Greece
This is a general talk in which we present some results on the theory of cones and geometry of Banach spaces. A special emphasis will be taken to some known but not so familiar results and problems on the bases for cones, isomorphic cones and cone characterization of Banach space properties.
A weighted Hardy inequality and nonexistence of positive solutions to some nonlinear problems
Abdelaziz Rhandi
Salerno, Italy
In this talk, we prove that the following weighted Hardy inequality (1)
\begin{equation} \left(\frac{|d-p|}{p}\right)^{p}\,\int_{\Omega}\,\frac{|u|^{p}}{|x|^{p}}\; d\mu \le \int_{\Omega}\,|\nabla u|^{p}\;d\mu + \left(\frac{|d-p|}{p}\right)^{p-1}\,\textrm{sgn}(d-p)\,\int_{\Omega}|u|^{p}\, \frac{(x^{t}Ax)^{p/2}}{|x|^{p}}\; d\mu \end{equation}holds with optimal Hardy constant $\left(\frac{|d-p|}{p}\right)^{p}$ for all $u\in W^{1,p}_{\mu,0}(\Omega)$ if the dimension $d \geq 2$, $1 < p < d$, and for all $u\in W^{1,p}_{\mu,0}(\Omega\setminus\{0\})$ if $p>d \geq 1$. Here we assume that $\Omega$ is an open subset of $\mathbb{R}^{d}$ with $0\in \Omega$, $A$ is a real $d\times d$-symmetric positive definite matrix, $c>0$, and
\[ d\mu: =\rho(x)\,dx \qquad\text{with density}\qquad \rho(x)=c\cdot\exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x\in \Omega . \]Due to the optimality of the Hardy constant in (1), we can establish nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential.
Banach limits
Evgeny Semenov
Voronezh, Russia
A linear functional $B\in l_\infty^*$ is called a Banach limit if
- $B\geq0$, i.e. $Bx \geq 0$ for $x \geq 0$ and $B1=1.$
- $B(Tx)=B(x)$ for all $x\in l_\infty$, where $T$ is a shift operator, i.e. \[T(x_1,x_2,\ldots)=(x_2,x_3,\ldots).\]
The existence of Banach limits was proven by S. Banach in his book. It follows from the definition, that $Bx=\lim_{n\to\infty}x_n$ for every convergent sequence $x\in l_\infty$ and $\|B\|_{l_\infty^*}=1.$ Denote the set of all Banach limits by $\mathfrak{B}.$ It is clear that $\mathfrak{B}$ is a closed convex subset of the unit sphere of the space $l_\infty^*.$ Hence, $\|B_1 - B_2\|\leq 2$ for every $B_1,B_2\in\mathfrak{B}.$
The set $A \subset l_\infty$ is called the set of uniqueness if the fact that two Banach limits $B_1$ and $B_2$ coincide on $A$ implies that $B_1=B_2$.
It was shown that under some restrictions on the operator $H$, acting on $l_\infty$, there exists such $B \in \mathfrak B$ that $Bx=BHx$ for every $x \in l_\infty$. We denote by $\mathfrak B(H)$ the set of all such Banach limits.
The sets of uniqueness, invariant Banach limits and extremal points of $\mathfrak{B}$ will be discussed in the talk.
Joint works with F.A. Sukochev and A.S. Usachev
Advances in modern noncommutative analysis
Fedor Sukochev
Sydney, Australia
Recent progress in noncommutative analysis has led to a resolution of three problems from the theory of spectral shift function (originated in 1940's in the works related to solid state theory). We discuss the resolution of a M.G. Krein's conjecture (1964), of a L.S. Koplienko's conjecture (1984) and of a conjecture due to F. Gesztesy, A. Pushnitski and B. Simon (2008).
Piling structure of families of matrix monotone functions and of matrix convex functions
Jun Tomiyama
Tokyo, Japan
Let $I$ be an interval on the real line and let $f$ be a real continuous function on $I$. Write $M_n$ for the $n\times n$ matrix algebra. The function $f$ is then said to be $n$-monotone if $f(a)\leq f(b)$, for all selfadjoint $a\leq b\in M_n$ with both spectra contained in $I$. It is $n$-convex if \(f(\lambda a + (1 - \lambda )b)\leq \lambda f(a) + (1 - \lambda)f(b)) \), for all selfadjoint $a,b\in M_n$ with both spectra contained in $I$, and all $0\leq\lambda\leq 1$.
The sequence $\{P_n(I)\}_{n=1}^\infty$ of sets of all $n$-monotone functions, and likewise the sequence $\{K_n(I)\}_{n=1}^\infty$ of sets of all $n$-convex functions, is decreasing with intersection $P_{\infty}(I)$, resp. $K_{\infty}(I)$. The functions belonging to these intersections are called operator monotone functions, resp. operator convex functions. One can think of all $P_n(I)$ as being ``piled'' on $P_\infty(I)$, and likewise for the convex case.
These notions were introduced and developed by K. Loewner and his two students O. Dobsch and F. Kraus in 1934-1936. Since then the theory has developed, with a great variety of applications to many fields of both pure and applied mathematics, and quite recently to quantum information theory.
As to the structure of the piles, it has been suggested in the literature that the inclusion $P_{n+1}(I)\subset P_n(I)$ must be proper, for all $n$, and likewise for the convex case. Yet concrete examples of functions in $P_{n}$, but not in $P_{n+1}$, are surprisingly lacking: even for $n=2$ only one example was known (G. Sparr, 1980). In this lecture, based on joint work with F. Hansen and G. Ji, we will provide an abundance of examples establishing the properness of all inclusions in the piling of the $P_n(I)$ and the $K_n(I)$.
Contributed Talks
Program
Adriaan Cornelis Zaanen
After a professorship in Delft from 1950-1956, Adriaan Cornelis Zaanen (1913-2003) held the chair of functional analysis in Leiden from 1956 until his retirement in 1982. He was one of the founding fathers of functional analysis in the Netherlands and was a prominent member of the international Positivity community. Several of his former PhD students are present at this conference.
A detailed description of his life and works can be found at the MacTutor History of Mathematics archive. For those who can read Dutch there is a still more detailed biography available in Nieuw Archief voor Wiskunde, the journal of the Royal Dutch Mathematical Society. With kind permission of the journal we post this paper, NAW 5/5, nr.1, maart 2004, 21--25. Those who cannot read Dutch may still appreciate the pictures.
Zaanen held his inaugural address in 1957. In Het Kleed der Wiskunde (Dutch) he comments on several aspects of mathematics in that period, amongst others on the in his opinion not always positive developments in mathematics in high school.