If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. In contrast, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected 'quantum Zeno subspace' is called 'quantum Zeno dynamics': for instance, if the measurements ascertain whether a quantum particle is in a given spatial region, the evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. We discuss the physical and mathematical aspects of this evolution, highlighting the open mathematical problems. We then analyze some alternative strategies to obtain a Zeno dynamics and show that they are physically equivalent.

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Quantum Zeno dynamics: mathematical and physical aspects

A Banach space X is said to have the Daugavet property if every operator T: X -* X of rank 1 satisfies 11 Id +Tl= 1 + flTIl. We show that then every weakly compact operator satisfies this equation as well and that X contains a copy of t1. However, X need not contain a copy of L1. We also study pairs of spaces X C Y and operators T: X -* Y satisfying I I J + T I I -_ 1-4- 1f T I I, where J: X -* Y is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with f Id +Tll 1 + Il T l I is as small as possible and give characterisations in terms of a smoothness condition.

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Banach spaces with the daugavet property

Cette derniere Newsletter avant les Grandes Vacances est, pur concours de circonstances, entierement consacree a l’Education. Mais l’Education sous l’angle de son importance pour la consecration dans l’excellence. La consecration, c’est ce qu’a obtenu Gisele Mophou-Massengo, quand son jury de soutenance l’a declaree Habilitee a Diriger des Recherches. C’est egalement la reconnaissance qui est accordee a Elina Marie Devoue par le jury du prix « Religions et Developpement » pour son article consacre aux Religions et a leurs impacts sur la societe dans la Caraibe.La reconnaissance du savoir et du savoir-faire ne passe pas forcement par sa consecration. Elle passe egalement par sa diffusion la plus large. C’est ce que tente de faire le Ceregmia en organisant depuis 2009 ses « seminaires intra Ceregmia » qui permettent l’expression de la transversalite du laboratoire sur des chantiers en cours ou deja finalises autour de thematiques que nous vous invitons a decouvrir en page 2. De meme, le Ceregmia a lance une nouvelle publication en 2010, ses « Documents de Travail ». Ces Documents d’audience plus confidentielle sont reellement reserves a un public de scientifiques. Diffuser le savoir en l’adaptant aux evolutions de l’environnement et aux exigences du marche. C’est ce qu’a fait le Ceregmia grâce a sa nouvelle offre de formation en Masters qui a ete habilitee par le Ministere et qui ouvrira des la rentree 2010 (p. 4). Par ailleurs, nous avions eu l’occasion de vous entretenir des changements intervenus a l’IFGCar suite au seisme du 12 janvier 2010 en Haiti. Nous sommes heureux aujourd’hui, de vous annoncer que l’IFGCar qui est desormais l’IAC retourne s’installer en Haiti. Les formations en Master reprendront des le 1 er . Octobre. Enfin, Nous vous signalons la parution de deux ouvrages aux Editions Ibis Rouge ils vous invitent a decouvrir la Guyane sous un angle moins connu, celui de l’anthropologie. Charley Granvorka

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Habilitation a diriger des recherches

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Recent Progress on the Daugavet Property

Geometric aspects of the Daugavet property.

Let X be a Banach space. We introduce a formal approach which seems to be useful in the study of those properties of operators on X which depend only on the norms of the images of elements. This approach is applied to the Daugavet equation for norms of operators; in particular we develop a general theory of narrow operators and rich subspaces of spaces X with the Daugavet property previously studied in the context of the classical spaces C(K) and L1( ).

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Narrow operators and rich subspaces of Banach spaces with the Daugavet property

We show that if T is a narrow operator (for the definition see below) on

 $$X = X_1 \oplus_1 X_2$$
 or

 $$X = X_1
\oplus_\infty X_2$$
 , then the restrictions to X1 and X2 are narrow and conversely. We also characterise by a version of the Daugavet property for positive operators on Banach lattices which unconditional sums of Banach spaces inherit the Daugavet property, and we study the Daugavet property for ultraproducts.

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Narrow operators and the Daugavet property for ultraproducts

The aim of this paper is to examine the convergence of Trotter’s product
formula when one of the C
 0-semigroups is replaced by a projection
(which can always be regarded as a constant degenerate semigroup). The
motivaton to study Trotter’s formula in this setting arises from the
fact that for ‘nice’ open sets $\Omega \subset \mathbb{R}^n$ the 
C
 0-semigroup on $L^{2}(\Omega)$ generated by the Laplacian with Dirichlet boundary
conditions can be obtained as a limit of a formula of this type.

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Trotter’s product formula for projections

We show that if $T$ is a narrow operator on $X=X_{1}\oplus_{1} X_{2}$ or $X=X_{1}\oplus_{\infty} X_{2}$, then the restrictions to $X_{1}$ and $X_{2}$ are narrow and conversely. We also characterise by a version of the Daugavet property for positive operators on Banach lattices which unconditional sums of Banach spaces inherit the Daugavet property, and we study the Daugavet property for ultraproducts.

Roman Shvidkoy

Papers

Banach spaces with the daugavet property

Narrow operators and rich subspaces of Banach spaces with the Daugavet property

Narrow operators and the Daugavet property for ultraproducts

Trotter’s product formula for projections

Narrow operators and the Daugavet property for ultraproducts