Weakly compact composition operators on spaces of Lipschitz functions

Let $X$ be a pointed compact metric space. Assuming that $\mathrm{lip}_0(X)$ has the uniform separation property, we prove that every weakly compact composition operator on spaces of Lipschitz functions $\mathrm{Lip}_0(X)$ and $\mathrm{lip}_0(X)$ is compact.Comment: 6 page

Algebraic reflexivity of diameter-preserving linear bijections between C(X)C(X)-spaces

We prove that if $X$ and $Y$ are first countable compact Hausdorff spaces, then the set of all diameter-preserving linear bijections from $C(X)$ to $C(Y)$ is algebraically reflexive.Comment: 13 page

Topological reflexivity of isometries on algebras of matrix-valued Lipschitz maps (Research on preserver problems on Banach algebras and related topics)

Let X and Y be compact metric spaces and let Mn(ℂ) be the Banach algebra of all n × n complex matrices. We prove that the set of all unital surjective linear isometries from Lip(X, Mn(ℂ)) to Lip(Y, Mn(ℂ)), whenever both spaces are endowed with the sum norm, is topologically reflexive

Compact Bloch mappings on the complex unit disc

The known duality of the space of Bloch complex-valued functions on the open complex unit disc $\mathbb{D}$ is addressed under a new approach with the introduction of the concepts of Bloch molecules and Bloch-free Banach space of $\mathbb{D}$. We introduce the notion of compact Bloch mapping from $\mathbb{D}$ to a complex Banach space and establish its main properties: invariance by M\"obius transformations, linearization from the Bloch-free Banach space of $\mathbb{D}$, factorization of their derivatives, inclusion properties, Banach ideal property and transposition on the Bloch function space. We state Bloch versions of the classical theorems of Schauder, Gantmacher and Davis-Figiel-Johnson-Pelczy\'nski.Comment: 25 page

Cohen strongly p-summing holomorphic mappings on Banach spaces

Let $E$ and $F$ be complex Banach spaces, $U$ be an open subset of $E$ and $1\leq p\leq\infty$. We introduce and study the notion of a Cohen strongly $p$-summing holomorphic mapping from $U$ to $F$, a holomorphic version of a strongly $p$-summing linear operator. For such mappings, we establish both Pietsch domination/factorization theorems and analyse their linearizations from $\mathcal{G}^\infty(U)$ (the canonical predual of $\mathcal{H}^\infty(U)$) and their transpositions on $\mathcal{H}^\infty(U)$. Concerning the space $\mathcal{D}_p^{\mathcal{H}^\infty}(U,F)$ formed by such mappings and endowed with a natural norm $d_p^{\mathcal{H}^\infty}$, we show that it is a regular Banach ideal of bounded holomorphic mappings generated by composition with the ideal of strongly $p$-summing linear operators. Moreover, we identify the space $(\mathcal{D}_p^{\mathcal{H}^\infty}(U,F^*),d_p^{\mathcal{H}^\infty})$ with the dual of the completion of tensor product space $\mathcal{G}^\infty(U)\otimes F$ endowed with the Chevet--Saphar norm $g_p$

Letter from the editor

Great challenges come for the world. Climatic change, clean energy, petroleum, politic points of view using science for their own benefit. During the last months, the catastrophes have been in the spotlight, not only for those who work on the sciences side to relieve the pain caused or avoid the consequences, but also those states' managers trying to reach political objectives as the earth stills moving, the water continues running through streets and housings; when the forest fires are not completely extinguished, some people is looking for economic or ideology tributes. It's part of our history. From the character Rhett Butler in Gone with the wind, the inspiring American novel, who made a vast fortune with the war, toward the contractors around the world drying public budgets as they explain the "risk management" concept. But science and politic are not going hand-in-hand.Recently, the Andean Region in South America has been the scenery of record-breaking floods and landslides; Peru was hit by heavy rains causing damages in the capital city Lima and the northern part of the country. In Colombia, the same heavy rains caused by El Niño climate cycle destroyed two neighborhoods in Mocoa, southern city capital of a department, state equivalent. Three hundreds of people died. At the same time, Bolivia is struggling with the drought consequences of climatic change. Country by country in this subcontinent the landscape is similar.  A political earthquake was the recent marches celebrated around the globe claiming for respect and support for the science. "We are marching today to remind people everywhere, our lawmakers especially, of the significance of science for our health and prosperity," said in Washington Bill Nye, a science educator with the fame of a rock star. People knows the solution for a better future is in the hands of our politicians; their decisions will allow the development or the constraint of the science. "Science, not silence," "Oceans are rising, and so are we," "Fund Science = Fun Future," "Policy without science is a Biohazard," "Science is not a partisan issue, it is the foundation of our society," "Science is NOT an opinion," were some of the messages exhibited by demonstrators in their marches.Humbly, from the Earth Sciences Research Journal, we want to adhere the same message: “It’s not magic, it’s science!” We present to you this new volume, is not so so much, but we hope every little bit helps