On the connected component of compact composition operators on the Hardy space

We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space $H^2$ on the unit disc. This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols. As a main tool we use Aleksandrov-Clark measures.Comment: 16 page

Sobre Ilex cubana Loes. (Aquifoliaceae) y especies afines

Se realiza la revisión taxonómica de cuatro especies del género Ilex L., distribuidas en el sector fitogeográfico Cuba-oriental según clasificación de Samek (1973). Se valoraron los resultados del análisis de numerosos materiales de herbario, así como estGonzález Gutiérrez, P. A. 2004. About Ilex cubana Loes. (Aquifoliaceae) and related species. Bot. Complut. 28: 93-100. The taxonomic revision of four species of Ilex L. spread in Eastern Cuba, according to the classification of Samek (1973), is carried out. Taking into account the results of morphological studies and filed's observations, Ilex turquinensis Alain, Ilex nunezii Borhidi and Ilex nannophylla Borhidi & Muñiz are considered heterotipic synonyms of Ilex cubana Loes. The description of this species is offered as well as data about its geographical distribution

Winnerless competition in coupled Lotka-Volterra maps

Winnerless competition is analyzed in coupled maps with discrete temporal evolution of the Lotka-Volterra type of arbitrary dimension. Necessary and sufficient conditions for the appearance of structurally stable heteroclinic cycles as a function of the model parameters are deduced. It is shown that under such conditions winnerless competition dynamics is fully exhibited. Based on these conditions different cases characterizing low, intermediate, and high dimensions are therefore computationally recreated. An analytical expression for the residence times valid in the N-dimensional case is deduced and successfully compared with the simulations.J.L.C. and E.D.G. acknowledge support from IVIC-141, L.A.G.-D. acknowledges support from IVIC-1089 and P.V. acknowledges support from MINECO TIN2012-30883

Band-diagonal operators on Banach lattices: matrix dynamics and invariant subspaces

We address the existence of non-trivial closed invariant ideals for positive operators defined on Banach lattices whose order is induced by an unconditional basis. In particular, for band-diagonal positive operators such existence is characterized whenever their matrix representations meet a positiveness criteria. For more general classes of positive operators, sufficient conditions are derived proving, particularly, the sharpness of such results from the standpoint of view of the matrix representations. The whole approach is based on studying the behavior of the dynamics of infinite matrices and the localization of the non-zero entries. Finally, we generalize a theorem of Grivaux regarding the existence of non-trivial closed invariant subspaces for positive tridiagonal operators to a more general class of band-diagonal operators showing, in particular, that a large subclass of them have non-trivial closed invariant subspaces but lack non-trivial closed invariant ideals.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEMinisterio de Ciencia e Innovacióninpres

C–H Insertion via Ruthenium Catalyzed gem-Hydrogenation of 1,3-Enynes

gem-Hydrogenation of an internal alkyne with the aid of [Cp*RuCl]4 as the precatalyst is a highly unorthodox transformation, in which one C atom of the triple bond is transformed into a methylene group, whereas the second C atom gets converted into a ruthenium carbene. In the case of 1,3-enynes bearing a propargylic steering substituent as the substrates, the reaction occurs regioselectively, giving rise to vinyl carbene complexes that adopt interconverting η1/η3-binding modes in solution; a prototypical example of such a reactive intermediate was characterized in detail by spectroscopic means. Although both forms are similarly stable, only the η3-vinyl carbene proved kinetically competent to insert into primary, secondary, or tertiary C–H bonds on the steering group itself or another suitably placed ether, acetal, orthoester, or (sulfon)amide substituent. The ensuing net hydrogenative C–H insertion reaction is highly enabling in that it gives ready access to spirocyclic as well as bridged ring systems of immediate relevance as building blocks for medicinal chemistry. Moreover, the reaction scales well and lends itself to the formation of partly or fully deuterated isotopologues. Labeling experiments in combination with PHIP NMR spectroscopy (PHIP = parahydrogen induced polarization) confirmed that the reactions are indeed triggered by gem-hydrogenation, whereas kinetic data provided valuable insights into the very nature of the turnover-limiting transition state of the actual C–H insertion step