Manuel Maestre

In the early nineties, R. M. Aron, B. Cole, T. W. Gamelin and W. B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of $$\ell _p$$ ℓ p ( $$1\le p<\infty $$ 1 ≤ p < ∞ ). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever $$p\in {\mathbb {N}}$$ p ∈ N ( $$p\ge 2$$ p ≥ 2 ), the fiber of every $$z\in B_{\ell _p}$$ z ∈ B ℓ p contains a set of cardinal $$2^{{\mathfrak {c}}}$$ 2 c such that any two elements of this set belong to different Gleason parts. For the case $$p=1$$ p = 1 , we complete the known description of the fibers, showing that, for each $$z\in \overline{B}_{\ell _1''}{\setminus } S_{\ell _1}$$ z ∈ B ¯ ℓ 1 ′ ′ \ S ℓ 1 , the fiber over z is not a singleton. Also, we establish that different fibers over elements in $$S_{\ell _1''}$$ S ℓ 1 ′ ′ cannot share Gleason parts.

This work is a thorough and detailed study on the geometry of the unit sphere of certain Banach spaces of homogeneous polynomials in ${\mathbb{R}}^2$. Specifically, we provide a complete description of the unit spheres, identify the extreme points of the unit balls, derive explicit formulas for the corresponding polynomial norms, and describe the techniques required to tackle these questions. To enhance the comprehensiveness of this work, we complement the results and their proofs with suitable diagrams and figures. The new results presented here settle some open questions posed in the past. For the sake of completeness of this work, we briefly discuss previous known results and provide directions of research and applications of our results.

We study the set $${\text {MA}}(X,Y)$$MA ( X , Y ) of operators between Banach spaces X and Y that attain their minimum norm, and the set $${\text {QMA}}(X,Y)$$QMA ( X , Y ) of operators that quasi attain their minimum norm. We characterize the Radon–Nikodym property in terms of operators that attain their minimum norm and obtain some related results about the density of the sets $${\text {MA}}(X,Y)$$MA ( X , Y ) and $${\text {QMA}}(X,Y)$$QMA ( X , Y ) . We show that every infinite-dimensional Banach space X has an isomorphic space Y, such that not every operator from X to Y quasi attains its minimum norm. We introduce and study Bishop–Phelps–Bollobás type properties for the minimum norm, including the ones already considered in the literature, and we exhibit a wide variety of results and examples, as well as exploring the relations between them.

We study the set $\operatorname{MA}(X,Y)$ of operators between Banach spaces $X$ and $Y$ that attain their minimum norm, and the set $\operatorname{QMA}(X,Y)$ of operators that quasi attain their minimum norm. We characterize the Radon-Nikodym property in terms of operators that attain their minimum norm and obtain some related results about the density of the sets $\operatorname{MA}(X,Y)$ and $\operatorname{QMA}(X,Y)$. We show that every infinite-dimensional Banach space $X$ has an isomorphic space $Y$ such that not every operator from $X$ to $Y$ quasi attains its minimum norm. We introduce and study Bishop-Phelps-Bollobás type properties for the minimum norm, including the ones already considered in the literature, and we exhibit a wide variety of results and examples, as well as exploring the relations between them.

Let $$B_N$$B N be the Euclidean ball of $${\mathbb {C}}^N$$C N . The space $$H^\infty (B_N)$$H ∞ ( B N ) of bounded holomorphic functions on $$B_N$$B N is known to have a predual, denoted by $$G^\infty (B_N)$$G ∞ ( B N ) . We study the functions in $$H^\infty (B_N)$$H ∞ ( B N ) that attain their norm as elements of the dual of $$G^\infty (B_N)$$G ∞ ( B N ) . We also examine similar questions for the polydisc algebra $$H^\infty ({\mathbb {D}}^N)$$H ∞ ( D N ) and for the space of Dirichlet series $$ {\mathcal {D}}^\infty ({\mathbb {C}}_+).$$D ∞ ( C + ) .

Let $X$ and $Y$ be complex Banach spaces with $B_X$ denoting the open unit ball of $X.$ This paper studies various aspects of the {\em holomorphic Lipschitz space} $\mathcal HL_0(B_X,Y)$, endowed with the Lipschitz norm. This space is the intersection of the spaces, $\operatorname{Lip}_0(B_X,Y)$ of Lipschitz mappings and $\mathcal H^\infty(B_X,Y)$ of bounded holomorphic mappings, from $B_X$ to $Y$. Thanks to the Dixmier-Ng theorem, $\mathcal HL_0(B_X, \mathbb C)$ is indeed a dual space, whose predual $\mathcal G_0(B_X)$ shares linearization properties with both the Lipschitz-free space and Dineen-Mujica predual of $\mathcal H^\infty(B_X)$. We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that $\mathcal G_0(B_X)$ contains a 1-complemented subspace isometric to $X$ and that $\mathcal G_0(X)$ has the (metric) approximation property whenever $X$ has it. We also analyze when $\mathcal G_0(B_X)$ is a subspace of $\mathcal G_0(B_Y)$, and we obtain an analogous to Godefroy's characterization of functionals with a unique norm preserving extension to the holomorphic Lipschitz context.