Mathematics and Computer Science

We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations P of the set {1,. .. , n}, there exists a closed infinite-dimensional Banach subspace of the space of n-linear forms on 1 such that, for all nonzero elements B of such a subspace, the Arens extension associated to the permutation σ of B is norm-attaining if and only if σ is an element of P. We also study the structure of the set of norm-attaining n-linear forms on c 0 .

We study norm attaining properties of the Arens extensions of multilinear forms defined on Banach spaces. Among other related results, we construct a multilinear form onℓ1with the property that only some fixed Arens extensions determined a priori attain their norms. We also study when multilinear forms can be approximated by ones with the property that only some of their Arens extensions attain their norms.

En esta propuesta se emplean conceptos de matemática avanzada adaptados a la enseñanza preuniversitaria, utilizando actividades intuitivas e inspiradoras. Se presenta la matemática como fruto del razonamiento lógico, fomentando el aprendizaje basado en la experimentación y focalizando la modelización a través de la experiencia física y la manipulación de objetos. Durante el desarrollo del trabajo se proponen tres talleres matemáticos (Anexos I, II y III) que pretenden cambiar la preconcepción que tienen los alumnos sobre la matemática como difícil e inaccesible. Para la realización de los talleres se usarán conceptos matemáticos de nivel superior como son: la teoría de grafos, la teoría de juego, la teoría de nudos, las demostración por inducción, la reducción al absurdo... This proposal uses advanced mathematical concepts adapted to high school education, using intuitive and inspiring activities. Mathematics is presented as a consequence of logical reasoning, promoting learning through experimentation and focusing on modelling through physical experience and manipulation of objects. During the development work three mathematical workshops are proposed (Annexes I, II and III) who try to change the preconception that the students have about mathematics as difficult and inaccessible. For the workshops will be used top-level mathematical concepts such as: graph theory, game theory, knot theory, the proof by induction, the reduction to absurdity...

Starting in 2007, the MFO publishes a preprint series which mainly contains research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs-Programme (RiP) and the Oberwolfach-Leibniz-Fellows (OWLF), but this can also include an Oberwolfach Lecture, for example. A preprint can have a size from 1-200 pages, and the MFO will publish it on its website as well as by hard copy. Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free hard copies (DIN A4, black and white copy) by surface mail. Of course, the full copy right is left to the authors. The MFO only needs the right to publish it on its website www.mfo.de as a documentation of the research work done at the MFO, which you are accepting by sending us your file. In case of interest, please send a pdf file of your preprint by email to or , respectively. The file should be sent to the MFO within 12 months after your stay as RiP or OWLF at the MFO. ri There are no requirements for the format of the preprint, except that the introduction should contain a short appreciation and that the paper size (respectively format) should be DIN A4, "letter" or "article". On the front page of the hard copies, which contains the logo of the MFO, title and authors, we shall add a running number (20XX-XX). We cordially invite the researchers within the RiP or OWLF programme to make use of this offer and would like to thank you in advance for your cooperation.

We consider A(), the Banach algebra of all functions f from = i∈I U i to C that are continuous on with respect to the product topology and separately holomorphic in , where I is an arbitrary set and U i are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off U i are uniformly dense in A(). This generalizes previous results where U i = D is the open unit disc in C or U i c is connected.

Given a proper holomorphic mapping g : Ω ⊆ C n −→ Ω ⊆ C n and an algebra of holomorphic functions B (e.g. P(K) where K ⊂ Ω is a compact set, H(U), A(U) or H ∞ (U) where U is an open and bounded set with U ⊂ Ω), we study the subalgebra B g of all functions compatible with the equivalence relation defined by the proper mapping g. We provide alternative representations of these algebras and describe the fibers in their spectra. Among other examples we relate the algebras of functions that are invariant under permutations and the algebras of functions defined on the symmetrized polydisk. Keywords Symmetric holomorphic functions • Fibers • Algebras of holomorphic functions • Several complex variables • Symmetrized polydisk

We show that the multiples of the backward shift operator on the spaces ℓ p , 1 ≤ p < ∞, or c 0 , when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of A-hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the Hadamard product, does not possess frequently hypercyclic algebras. On the other hand, we show that for any frequently hypercyclic operator T on any Banach space, F HC(T) is algebrable for a suitable product, and in some cases it is even strongly algebrable.

We show that if U is an arbitrary open subset of a Riemann surface and ϕ an arbitrary continuous function on the boundary ∂U , then there exists a holomorphic functionφ on U such that, for every p ∈ ∂U ,φ(x) → ϕ(p), as x → p outside a set of density 0 at p relative to U. These "solutions to a boundary problem" are not unique. In fact they can be required to have interpolating properties and also to assume all complex values near every boundary point. Our result is new even for the unit disc. c

We show that for any Banach space and any compact topological group G ⊂ L ( X ) G\subset L(X) such that the norm of X X is G G -invariant, the set of norm attaining G G -invariant functionals on X X is dense in the set of all G G -invariant functionals on X X , where a mapping f f is called G G -invariant if for every x ∈ X x\in X and every g ∈ G g\in G , f ( g ( x ) ) = f ( x ) f\big (g(x)\big )=f(x) . In contrast, we show also that the analog of Bollobás result does not hold in general. A version of Bollobás and James’ theorems is also presented.

We introduce and investigate the mth polarization constant of a Banach space X for the numerical radius. We first show the difference between this constant and the original mth polarization constant associated with the norm by proving that the new constant is minimal if and only if X is strictly convex, and that there exists a Banach space which does not have an almost isometric copy of 2 1 , such that the second polarization constant for the numerical radius is maximal. We also give a negative answer for complex Banach spaces to the question of Choi and Kim (J Lond Math Soc (2) 54(1):135-147, 1996) whether m k=1 k m (m k) m! is the optimal upper bound for the mth polarization constant for arbitrary m ∈ N. Finally, we generalize the result of García et al.

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on Fréchet sequence spaces that are algebras when endowed with coordinatewise multiplication or with the Cauchy product. As a particular case we obtain that the sets of hypercyclic vectors for Rolewicz's and MacLane's operators are algebrable.

We use approximation in measure to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann hypothesis are suggested.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$ there is a function f holomorphic in the unit disc, having $\psi $ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions f of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.

In this paper, we study geometric properties of the set of group invariant continuous linear functionals and operators between Banach spaces. In particular, we present group invariant versions of the Hahn-Banach separation theorem and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators, establishing group invariant versions of the properties α of Schachermayer and β of Lindenstrauss, and presenting relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain’s result, establishing that if X has the Radon-Nikodým property, then X has the G-Bishop-Phelps theorem for G-invariant operators whenever G is a compact group.

We consider A(Ω), the Banach space of functions f from Ω¯¯¯¯=∏i∈IUi¯¯¯¯¯ to C that are continuous with respect to the product topology and separately holomorphic, where I is an arbitrary set and Ui are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off Ui¯¯¯¯¯ are uniformly dense in A(Ω). This generalizes previous results where Ui=D is the open unit disc in C or Ui¯¯¯¯¯c is connected.

We study the group invariant continuous polynomials on a Banach space X that separate a given set K in X and a point z outside K. We show that if X is a real Banach space, G is a compact group of L(X), K is a G-invariant set in X, and z is a point outside K that can be separated from K by a continuous polynomial Q, then z can also be separated from K by a G-invariant continuous polynomial P. It turns out that this result does not hold when X is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that X has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group invariant polynomial from the closed unit ball.