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

1. $B\geq0$, i.e. $Bx \geq 0$ for $x \geq 0$ and $B1=1.$
2. $B(Tx)=B(x)$ 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.