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

The Bishop-Phelps-Bollobás Property for Weighted Holomorphic Mappings

Given an open subset U of a complex Banach space E, a weight v on U and a complex Banach space F, let Hv∞(U,F) denote the Banach space of all weighted holomorphic mappings from U into F, endowed with the weighted supremum norm. We introduce and study a version of the Bishop–Phelps–Bollobás property for Hv∞(U,F) (WH∞-BPB property, for short). A result of Lindenstrauss type with sufficient conditions for Hv∞(U,F) to have the WH∞-BPB property for every space F is stated. This is the case of Hvp∞(D,F) with p≥1, where vp is the standard polynomial weight on D. The study of the relations of the WH∞-BPB property for the complex and vector-valued cases is also addressed as well as the extension of the cited property for mappings f∈Hv∞(U,F) such that vf has a relatively compact range in F

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 $g_p$