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In this paper, we study the (uniform) strong subdifferentiability of the norms of the Banach spaces $\mathcal{P}(^N X, Y^*)$, $X \hat{\otimes}_\pi \cdots \hat{\otimes}_\pi X$ and $\hat{\otimes}_{\pi_s,N} X$. Among other results, we characterize when the norms of the spaces $\mathcal{P}(^N \ell_p, \ell_{q}), \mathcal{P}(^N l_{M_1}, l_{M_2})$, and $\mathcal{P}(^N d(w,p), l_{M_2})$ are strongly subdifferentiable. Analogous results for multilinear mappings are also obtained. Since strong subdifferentiability of a dual space implies reflexivity, we improve some known results on the reflexivity of spaces of $N$-homogeneous polynomials and $N$-linear mappings. Concerning the projective (symmetric) tensor norms, we provide positive results on the subsets $U$ and $U_s$ of elementary tensors on the unit spheres of $X \hat{\otimes}_\pi \cdots \hat{\otimes}_\pi X$ and $\hat{\otimes}_{\pi_s,N} X$, respectively. Specifically, we prove that $\hat{\otimes}_{\pi_s,N} \ell_2$ and $\ell_2 \hat{\otimes}_\pi \cdots \hat{\otimes}_\pi \ell_2$ are uniformly strongly subdifferentiable on $U_s$ and $U$, respectively, and that $c_0 \hat{\otimes}_{\pi_s} c_0$ and $c_0 \hat{\otimes}_\pi c_0$ are strongly subdifferentiable on $U_s$ and $U$, respectively, in the complex case., Comment: 38 pages

We study holomorphic functions attaining weighted norms and its connections with the classical theory of norm attaining holomorphic functions. We prove that there are polynomials on $\ell_p$ which attain their weighted but not their supremum norm and viceversa. Nevertheless, we also prove that in the context of polynomials of fixed degree both norms are in fact equivalent. This leads us to the main problem of the paper, namely, whether the holomorphic functions attaining their weighted norm are dense. Although we exhibit an example where this does not hold, as the main theorem of our paper, we prove the denseness provided the domain space is uniformly convex. In fact, we provide a Bollob\'as type theorem in this setting. For the proof of such a result we develop a new geometric technique.

For $1\leq p1$ or $\Gamma$ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in $\ell_1(\Gamma)$ admits a $C^1$-smooth norm.

Let $\mathcal{X}$ be a Banach space with a fundamental biorthogonal system and let $\mathcal{Y}$ be the dense subspace spanned by the vectors of the system. We prove that $\mathcal{Y}$ admits a $C^\infty$-smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that $\mathcal{Y}$ admits locally finite, $\sigma$-uniformly discrete $C^\infty$-smooth and LFC partitions of unity and a $C^1$-smooth LUR norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every WLD Banach space (hence, all reflexive ones), $L_1(\mu)$ for every measure $\mu$, $\ell_\infty(\Gamma)$ spaces for every set $\Gamma$, $C(K)$ spaces where $K$ is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum {\sf MM}, all Banach spaces of density $\omega_1$ are covered by our result., Comment: Int. Math. Res. Not. IMRN (online first)

For $1\leq p1$ or $Γ$ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in $\ell_1(Γ)$ admits a $C^1$-smooth norm.

In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn-Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators; we establish group invariant versions of the properties $\alpha$ of Schachermayer and $\beta$ of Lindenstrauss, and present relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain's result, which says that if $X$ has the Radon-Nikod\'ym property, then $X$ has the $G$-Bishop-Phelps property for $G$-invariant operators whenever $G \subseteq \mathcal{L}(X)$ is a compact group of isometries on $X$., Comment: 28 pages

In this paper, we are interested in giving two characterizations for the so-called property {\bf L}$_{o,o}$, a local vector valued Bollob\'as type theorem. We say that $(X, Y)$ has this property whenever given $\eps > 0$ and an operador $T: X \rightarrow Y$, there is $\eta = \eta(\eps, T)$ such that if $x$ satisfies $\|T(x)\| > 1 - \eta$, then there exists $x_0 \in S_X$ such that $x_0 \approx x$ and $T$ itself attains its norm at $x_0$. This can be seen as a strong (although local) Bollob\'as theorem for operators. We prove that the pair $(X, Y)$ has the {\bf L}$_{o,o}$ for compact operators if and only if so does $(X, \K)$ for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when $(X \pten Y, \K)$ satisfies the {\bf L}$_{o,o}$ for linear functionals under strict convexity or Kadec-Klee property assumptions in one of the spaces. As a consequence, we generalize some results in the literature related to the strongly subdifferentiability of the projective tensor product and show that $(L_p(\mu) \times L_q(\nu); \K)$ cannot satisfy the {\bf L}$_{o,o}$ for bilinear forms., Comment: 10 pages

In this paper, we are interested in giving two characterizations for the so-called property {\bf L}$_{o,o}$, a local vector valued Bollob��s type theorem. We say that $(X, Y)$ has this property whenever given $\eps > 0$ and an operador $T: X \rightarrow Y$, there is $��= ��(\eps, T)$ such that if $x$ satisfies $\|T(x)\| > 1 - ��$, then there exists $x_0 \in S_X$ such that $x_0 \approx x$ and $T$ itself attains its norm at $x_0$. This can be seen as a strong (although local) Bollob��s theorem for operators. We prove that the pair $(X, Y)$ has the {\bf L}$_{o,o}$ for compact operators if and only if so does $(X, \K)$ for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when $(X \pten Y, \K)$ satisfies the {\bf L}$_{o,o}$ for linear functionals under strict convexity or Kadec-Klee property assumptions in one of the spaces. As a consequence, we generalize some results in the literature related to the strongly subdifferentiability of the projective tensor product and show that $(L_p(��) \times L_q(��); \K)$ cannot satisfy the {\bf L}$_{o,o}$ for bilinear forms., 10 pages

In this paper, we are interested in studying the set $\mathcal{A}_{\|\cdot\|}(X, Y)$ of all norm-attaining operators $T$ from $X$ into $Y$ satisfying the following: given $\epsilon>0$, there exists $\eta$ such that if $\|Tx\| > 1 - \eta$, then there is $x_0$ such that $\| x_0 - x\| < \epsilon$ and $T$ itself attains its norm at $x_0$. We show that every norm one functional on $c_0$ which attains its norm belongs to $\mathcal{A}_{\|\cdot\|}(c_0, \mathbb{K})$. Also, we prove that the analogous result holds neither for $\mathcal{A}_{\|\cdot\|}(\ell_1, \mathbb{K})$ nor $\mathcal{A}_{\|\cdot\|}(\ell_{\infty}, \mathbb{K})$. Under some assumptions, we show that the sphere of the compact operators belongs to $\mathcal{A}_{\|\cdot\|}(X, Y)$ and that this is no longer true when some of these hypotheses are dropped. The analogous set $\mathcal{A}_{nu}(X)$ for numerical radius of an operator instead of its norm is also defined and studied. We present a complete characterization for the diagonal operators which belong to the sets $\mathcal{A}_{\| \cdot \|}(X, X)$ and $\mathcal{A}_{nu}(X)$ when $X=c_0$ or $\ell_p$. As a consequence, we get that the canonical projections $P_N$ on these spaces belong to our sets. We give examples of operators on infinite dimensional Banach spaces which belong to $\mathcal{A}_{\| \cdot \|}(X, X)$ but not to $\mathcal{A}_{nu}(X)$ and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums., Comment: In this version, we have added a complete characterization for the sets $\mathcal{A}_{\|\cdot\|}$ and $\mathcal{A}_{nu}$ for the diagonal operators on $c_0$ and $\ell_p$

We continue a line of study about some local versions of Bishop-Phelps-Bollob\'as type properties for bounded linear operators. We introduce and focus our attention on two of these local properties, which we call L$_{p, o}$ and L$_{o, p}$, and we explore the relation between them and some geometric properties of the underlying spaces, such as spaces having strict convexity, local uniform rotundity, and property $\beta$ of Lindenstrauss. At the end of the paper, we present a diagram comparing all the existing Bishop-Phelps-Bollob\'as type properties with each other. Some open questions are left throughout the article.

It has been recently presented some local versions of the Bishop-Phelps-Bollob��s type property for operators. In the present article, we continue studying these properties for multilinear mappings. We show some differences between the local and uniform versions of the Bishop-Phelps-Bollob��s type results for multilinear mappings, and also provide some interesting examples which shows that this study is not just a mere generalization of the linear case. We study those properties for bilinear forms on $\ell_p \times \ell_q$ using the strong subdifferentiability of the norm of the Banach space $\ell_p \hat{\otimes}_�� \ell_{q}$. Moreover, we present necessary and sufficient conditions for the norm of a Banach space $Y$ to be strongly subdifferentiable through the study of these properties for bilinear mappings on $\ell_1^N \times Y$.

We continue a line of study about some local versions of Bishop-Phelps-Bollob��s type properties for bounded linear operators. We introduce and focus our attention on two of these local properties, which we call L$_{p, o}$ and L$_{o, p}$, and we explore the relation between them and some geometric properties of the underlying spaces, such as spaces having strict convexity, local uniform rotundity, and property $��$ of Lindenstrauss. At the end of the paper, we present a diagram comparing all the existing Bishop-Phelps-Bollob��s type properties with each other. Some open questions are left throughout the article.

In this paper, we are interested in studying the set $\mathcal{A}_{\|\cdot\|}(X, Y)$ of all norm-attaining operators $T$ from $X$ into $Y$ satisfying the following: given $��>0$, there exists $��$ such that if $\|Tx\| > 1 - ��$, then there is $x_0$ such that $\| x_0 - x\| < ��$ and $T$ itself attains its norm at $x_0$. We show that every norm one functional on $c_0$ which attains its norm belongs to $\mathcal{A}_{\|\cdot\|}(c_0, \mathbb{K})$. Also, we prove that the analogous result holds neither for $\mathcal{A}_{\|\cdot\|}(\ell_1, \mathbb{K})$ nor $\mathcal{A}_{\|\cdot\|}(\ell_{\infty}, \mathbb{K})$. Under some assumptions, we show that the sphere of the compact operators belongs to $\mathcal{A}_{\|\cdot\|}(X, Y)$ and that this is no longer true when some of these hypotheses are dropped. The analogous set $\mathcal{A}_{nu}(X)$ for numerical radius of an operator instead of its norm is also defined and studied. We present a complete characterization for the diagonal operators which belong to the sets $\mathcal{A}_{\| \cdot \|}(X, X)$ and $\mathcal{A}_{nu}(X)$ when $X=c_0$ or $\ell_p$. As a consequence, we get that the canonical projections $P_N$ on these spaces belong to our sets. We give examples of operators on infinite dimensional Banach spaces which belong to $\mathcal{A}_{\| \cdot \|}(X, X)$ but not to $\mathcal{A}_{nu}(X)$ and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums., In this version, we have added a complete characterization for the sets $\mathcal{A}_{\|\cdot\|}$ and $\mathcal{A}_{nu}$ for the diagonal operators on $c_0$ and $\ell_p$

In this paper we consider a stronger property than the Bishop-Phelps-Bollob\'{a}s property for various classes of operators on a complex Hilbert space. The Bishop-Phelps-Bollob\'as {\it point} property for some class $\mathcal{A} \subset \mathcal{L}(H)$ says that if one starts with a norm one operator $T$ belonging to $\mathcal{A}$, which almost attains its norm at some norm one vector $x_0$, then there is a new operator $S$, belonging to the same class $\mathcal{A}$, which is close to $T$ and attains its norm at the same vector $x_0$. We study it for classical operators on a complex Hilbert spaces such as self-adjoint, anti-symmetric, unitary, compact, normal, and Schatten-von Neumann operators. We also solve analogous problems by replacing the norm of an operator by its numerical radius., Comment: 14 pages

In this paper we consider a stronger property than the Bishop-Phelps-Bollob��s property for various classes of operators on a complex Hilbert space. The Bishop-Phelps-Bollob��s {\it point} property for some class $\mathcal{A} \subset \mathcal{L}(H)$ says that if one starts with a norm one operator $T$ belonging to $\mathcal{A}$, which almost attains its norm at some norm one vector $x_0$, then there is a new operator $S$, belonging to the same class $\mathcal{A}$, which is close to $T$ and attains its norm at the same vector $x_0$. We study it for classical operators on a complex Hilbert spaces such as self-adjoint, anti-symmetric, unitary, compact, normal, and Schatten-von Neumann operators. We also solve analogous problems by replacing the norm of an operator by its numerical radius., 14 pages

In this paper we study conditions assuring that the Bishop-Phelps-Bollob��s property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair (X, Y) of Banach spaces having the BPBp, (a) if Y1 is an absolute summand of Y, then (X, Y1) has the BPBp; (b) if X1 is an absolute summand of X of type 1 or \infty, then (X1, Y) has the BPBp. Besides, analogous results for the BPBp for compact operators and for the density of norm attaining operators are also given. We also show that the Bishop-Phelps-Bollob��s property for numerical radius is inherited by absolute summands of type 1 or \infty. Moreover, we provide analogous results for numerical radius attaining operators and for the BPBp for numerical radius for compact operators.

Given two real Banach spaces $X$ and $Y$ with dimensions greater than one, it is shown that there is a sequence $\{T_n\}_{n\in \mathbb{N}}$ of norm attaining norm-one operators from $X$ to $Y$ and a point $x_0\in X$ with $\|x_0\|=1$, such that $\|T_n(x_0)\|\longrightarrow 1$ but $\inf_{n \in \mathbb{N}} \{\mbox{dist} (x_0,\,\{x\in X: \|T_n(x)\|=\|x\|=1\})\} >0.$ This shows that a version of the Bishop-Phelps-Bollob��s property in which the operator is not changed is possible only if one of the involved Banach spaces is one-dimensional., The content of this paper overlaps with the old version of arXiv:1709.00032 (arXiv:1709.00032v1, submitted on 31 Aug 2017). Nevertheless, there is no intersection between the present version and the updated one of arXiv:1709.00032 (arXiv:1709.00032v2, submitted on 26 Sep 2018)

The main aim of this paper is to prove a Bishop-Phelps-Bollob\'as type theorem on the unital uniform algebra A_{w^*u}(B_{X^*}) consisting of all w^*-uniformly continuous functions on the closed unit ball B_{X^*} which are holomorphic on the interior of B_{X^*}. We show that this result holds for A_{w^*u}(B_{X^*}) if X^* is uniformly convex or X^* is the uniformly complex convex dual space of an order continuous absolute normed space. The vector-valued case is also studied.

Tese arquivada ao abrigo da Portaria nº 227/2017 de 25 de julho

The main aim of this paper is to prove a Bishop-Phelps-Bollob��s type theorem on the unital uniform algebra A_{w^*u}(B_{X^*}) consisting of all w^*-uniformly continuous functions on the closed unit ball B_{X^*} which are holomorphic on the interior of B_{X^*}. We show that this result holds for A_{w^*u}(B_{X^*}) if X^* is uniformly convex or X^* is the uniformly complex convex dual space of an order continuous absolute normed space. The vector-valued case is also studied.

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X, Y)$ has the pointwise Bishop-Phelps-Bollob��s property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X, Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X, Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_p(��)$ spaces fail to have this property when $p>2$. For universal pointwise BPB range space, we show that every simultaneously uniformly convex and uniformly smooth Banach space fails it if its dimension is greater than one. We also discuss a version of the pointwise BPB property for compact operators., 19 pages, to appear in the Canadian J. Math. In this version, section 6 and the appendix of the previous version have been removed

This dissertation is devoted to the study of the Bishop-Phelps-Bollobás property in different contexts. In Chapter 1 we give a historical resume and the motivation behind this property as the classics Bishop-Phelps and Bishop-Phelps-Bollobás theorems. We define the Bishop-Phelps-Bollobás property (BPBp) and we comment on some important current results. In Chapter 2 we study similar properties to the BPBp. First, we define the Bishop-Phelps-Bollobás point property (BPBpp). The BPBpp is stronger than the BPBp. We study it for bounded linear operators and then for bilinear mappings. After that, we study two more similar properties: properties 1 and 2. Property 2 is just the dual property of the BPBpp. We observe that is not so easy to get positive results for this property and we comment some differences between this property and the BPBpp since, although they are very similar at first sight, they have completely different behavior from each other. On the other hand, property 1, which is defined similarly but depending on a fixed norm one bounded linear operator, has some positive examples. We finish this chapter studying the BPBpp version for numerical radius on complex Hilbert spaces and the BPBp for absolute norms. We dedicate Chapter 3 to the study of the BPBp for compact operators which is defined analogously to the BPBp but now considering just this type of operator. Our strategy is to study the conditions that the Banach spaces must satisfy to get a BPBp for compact operators and use technical results to pass the BPBp for compact operators from sequences to functions spaces. We apply these results for both domain and range situations. For example, if the pair (c_0; Y) has the property so does the pair (C_0(L); Y) for every locally compact Hausdorff topological space L. We also prove that we can pass the BPBp for compact operators from the pair (X; \ell_p(Y)) to the pair (X; L_p(\mu,Y)). Moreover, if Y has a certain geometric property, then the pairs (X; L_{\infty} (\nu,Y)) and (X; C(K,Y)) have the Bishop-Phelps-Bollobás property for compact operators. In the last chapter the BPBp is extended to the multilinear version. We discuss when it is possible to pass some known results about the BPBp for operators to the multilinear case. We give some results for symmetric multilinear mappings and homogeneous polynomials. Still on this chapter, we study the numerical radius on the set of all multilinear mappings defined in L_1. We prove that, in this case, the numerical radius and the norm of a multilinear mapping coincide. We also study the BPBp for numerical radius (BPBp-nu) for multilinear mappings. It is shown that if X is finite dimensional, then X satisfies this property. On the other hand, L_1 fails it although L_1 has the analogously property for bounded linear operators. We also prove that if a c_0 or a \ell_1-sum satisfies it, then each component of the direct sum also satisfies the BPBp-nu for multilinear mappings. We finish the dissertation by presenting a list of open problems with the intention to expand new horizons. Also we present tables which summary the pairs of classical Banach spaces satisfying the Bishop-Phelps-Bollobás property with the purpose to put the reader in the current scenario on this topic.

In this article, we study a version of the Bishop-Phelps-Bollob\'as property. We investigate a pair of Banach spaces $(X, Y)$ such that every operator from $X$ into $Y$ is approximated by operators which attains its norm at the same point where the original operator almost attains its norm. In this case, we say that such a pair has the Bishop-Phelps-Bollob\'as point property (BPBpp). We characterize uniform smoothness in terms of BPBpp and we give some examples of pairs $(X, Y)$ which have and fail this property. Some stability results are obtained about $\ell_1$ and $\ell_\infty$ sums of Banach spaces and we also study this property for bilinear mappings.

In this paper we introduce the strong Bishop-Phelps-Bollob\'as property (sBPBp) for bounded linear operators between two Banach spaces $X$ and $Y$. This property is motivated by a Kim-Lee result which states, under our notation, that a Banach space $X$ is uniformly convex if and only if the pair $(X,\mathbb{K})$ satisfies the sBPBp. Positive results of pairs of Banach spaces $(X,Y)$ satisfying this property are given and concrete pairs of Banach spaces $(X, Y)$ failing it are exhibited. A complete characterization of the sBPBp for the pairs $(\ell_p, \ell_q)$ is also provided., Comment: 17 pages, 1 figure

We study the Bishop-Phelps-Bollob\'as property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikod\'ym property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$ for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any positive measure $\nu$ and $1< p< \infty$. For $1\leqslant p, Comment: 21 pages

In this article, we study a version of the Bishop-Phelps-Bollob��s property. We investigate a pair of Banach spaces $(X, Y)$ such that every operator from $X$ into $Y$ is approximated by operators which attains its norm at the same point where the original operator almost attains its norm. In this case, we say that such a pair has the Bishop-Phelps-Bollob��s point property (BPBpp). We characterize uniform smoothness in terms of BPBpp and we give some examples of pairs $(X, Y)$ which have and fail this property. Some stability results are obtained about $\ell_1$ and $\ell_\infty$ sums of Banach spaces and we also study this property for bilinear mappings.

We study the Bishop-Phelps-Bollob��s property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikod��m property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then so does $(L_1(��,X),Y)$ for every positive measure $��$; as a consequence, $(L_1(��,X),Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(��)$ for any positive measure $��$ and $1< p< \infty$. For $1\leqslant p, 21 pages

In this paper we introduce the strong Bishop-Phelps-Bollob��s property (sBPBp) for bounded linear operators between two Banach spaces $X$ and $Y$. This property is motivated by a Kim-Lee result which states, under our notation, that a Banach space $X$ is uniformly convex if and only if the pair $(X,\mathbb{K})$ satisfies the sBPBp. Positive results of pairs of Banach spaces $(X,Y)$ satisfying this property are given and concrete pairs of Banach spaces $(X, Y)$ failing it are exhibited. A complete characterization of the sBPBp for the pairs $(\ell_p, \ell_q)$ is also provided., 17 pages, 1 figure