## General

The seventh Positivity conference was 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 was 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 consisted 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:

- 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)

For inquiries, or inclusion in the mailing list for this conference, please write to positivity2013@gmail.com.

## Announcements

### July 31st, 2013. Proceedings and photos

Dear participant,

Thank you for attending Positivity VII. This email contains some additional information concerning the proceedings and posting photos.

#### Proceedings

The proceedings of this conference, edited by Marcel de Jeu, Ben de Pagter, Onno van Gaans, and Mark Veraar, will be published by Birkhäuser as a book in their Trends in Mathematics series. All participants can submit, and both research papers and survey papers are welcome. The contributions will be subject to a serious refereeing procedure.

In case of acceptance, the authors are responsible for converting their file to the style as required by Birkhäuser. Details about style files etc. will be supplied later.

The deadline for submission is **1 February, 2014**, and submission is possible as of today. For all communication concerning the proceedings, including submission, the conference's email address positivity2013@gmail.com will be used.

Future additional information will be made available through the conference's website http://websites.math.leidenuniv.nl/positivity2013/ , and an email will be sent to all participants when new information has been posted.

#### Photos

There are already some photos on the website http://websites.math.leidenuniv.nl/positivity2013/ , and we welcome more photos by the participants. You can send them by email to positivity2013@gmail.com, preferably before 1 September, or, if the amount of data is too large, use the free file transfer service provided by WeTransfer ). This service is easy to use. Once you have accepted the general conditions, you will see a small window, and the first step is to select the files you want to send us. The second is to type positivity2013@gmail.com as the receiving address, and the third is to type your own address. In the message field, please mention your name, as an email address is not always clear and you might be using another address than we have. If you then click the Send button, the system will do the rest automatically, upload your files and send us a message that we can download them.

With kind regards,

the organizers.

### July 31st, 2013. Permission to post presentation

Dear participant,

In order to enhance the dissemination of the results presented during the conference, we would like to post the presentation as you may have given on the conference's website. If you gave a lecture, and if you agree to this, could you send us your permission by responding to this message, preferably before 1 September?

If you consent, we will, in principle, post the file as we have it. We think we have all the files and if not we will contact you.

With kind regards,

the organizers.

### July 13th, 2013. Program and practical information Positivity VII (22-26 July, 2013)

Dear participant,

On the website http://websites.math.leidenuniv.nl/positivity2013/ you will now find the program and some practical information, including travel information to Leiden and to the conference venue. The latter is the dish-like lecture room building of the Gorlaeus Laboratoria at the Science Campus of Leiden University, and it is located at

Einsteinweg 55

2333 CC Leiden

The Netherlands.

Registration with coffee is on Monday 22 July from 8:30-9:45, and after the opening of the conference and some practical announcements from 9:45-10:00 the first lecture will start at 10:00.

Positivity VII will have approximately 170 participants. You are kindly requested to arrive well before 9:45, so that we can handle all registrations properly before the opening.

With 16 plenary and 122 contributed lectures we have a tight schedule. To make it run as smoothly as possible, you are kindly requested to transfer any PDF you may want to use onto the permanent PC with which each of the lecture rooms is equipped before the start of the session. Someone will be available during the session in case technical assistance is needed, and this person will also be present 15 minutes before the start of the session to transfer the file and to answer any questions you may have about the (easy to use) equipment.

We look forward to seeing you soon in Leiden.

With kind regards,

the organizers.

### 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).

## 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*:

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:

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 $K_n(I)$.

### $\gamma$-Radonifying operators

### Jan van Neerven

#### Delft, The Netherlands

In recent years, $\gamma$-radonifying operators have become a mainstream tool in various branches of vector-valued Analysis. Roughly speaking, they give the correct generalisation to the Banach space valued context of the notion of a square integrable function and thus permit the extension to Banach spaces of various $L^2$-isometries, such as the Fourier-Plancherel isometry and the It\^o isometry. In this talk we aim to present a birds eye view of the theory of $\gamma$-radonifying operators and to point out some of its most salient applications.## Contributed Talks

## Program

The program booklet was part of the conference handout. Modified so as to reflect the program and list of participants as they developed during the week, it can be downloaded here.

## Presentations

Presentations for which the participants have agreed that they could be posted are available here.

## Proceedings

The proceedings of this conference, edited by Marcel de Jeu, Ben de Pagter, Onno van Gaans, and Mark Veraar, will be published by Birkhäuser as a book in their Trends in Mathematics series. All participants can submit, and both research papers and survey papers are welcome. The contributions will be subject to a serious refereeing procedure.

The contract terms foresee one free copy of the proceedings for the first-mentioned (or the corresponding) author of each contribution.

In case of acceptance, the authors are responsible for converting their file to the style as required by Birkhäuser. The ZIP-file containing the self-explanatory files is here. It is not necessary to use this style already in a submitted paper.

The deadline for submission is **1 February, 2014**. For all communication concerning the proceedings, including submission, the conference's email address positivity2013@gmail.com will be used.

Future additional information will be made available through the conference's website http://websites.math.leidenuniv.nl/positivity2013/, and an email will be sent to all participants when new information has been posted.

## Photos

If you took photos at the conference that you would like to share, please send them using WeTransfer (to the conference email address "positivity2013@gmail.com") before **Friday, August 30th, 2013.** Please send only photos depicting participants.

In a download of all collected photos will be made available in due time.

## 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.