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

In this paper, we introduce two Bishop-Phelps-Bollobás type properties for bounded linear operators between two Banach spaces X and Y: property 1 and property 2. These properties are 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,\K) satisfies property 2. Positive results for pairs of Banach spaces (X, Y) satisfying property 1 are given and concrete pairs of Banach spaces (X, Y) failing both properties are exhibited. A complete characterization for property 1 of the pairs (\ell_p, \ell_q) is also provided.

Here, we study the Bishop-Phelps-Bollobás property (BPBp) for compact operators. We present some abstract techniques that allow us to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let X and 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-Nikodým 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 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 \leq p < \infty, if (X, \ell_p(Y)) has the BPBp for compact operators, then so does (X, L_p(\mu, Y)) for every positive measure \mu such that L_1(\mu) is infinite-dimensional. If (X, Y) has the BPBp for compact operators, then so do (X, L_{\infty} (\mu, Y)) for every \sigma-finite positive measure \mu and (X, C(K, Y)) for every compact Hausdorff topological space K.

We study the Bishop-Phelps-Bollobás property and the Bishop-Phelps-Bollobás property for numerical radius. Our main aim is to extend some known results about norm or numerical radius attaining operators to the multilinear and polynomial cases. We characterize the pair (\ell_1(X), Y) to have the BPBp for bilinear forms and prove that on L_1(\mu) the numerical radius and the norm of a multilinear mapping are the same. We also show that L_1(\mu) fails the BPBp-nu for multilinear mappings although L_1(\mu) satisfies it in the operator case for every measure \mu.

The main aim of this paper is to prove a Bishop-Phelps-Bollobás type theorem on the unital uniform algebra \mathcal{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 \mathcal{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. Throughout the paper, we consider complex Banach spaces.

In this paper, we introduce some local versions of Bishop-Phelps-Bollobás type property for operators. That is, the function \eta which appears in their definitions depends not only on a given \eps >0, but also on either a fixed norm-one operator T or a fixed norm-one vector x. We investigate those properties and show differences between local and uniform versions.

We study approximations 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-Bollobás 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(\mu)-spaces fail to have this property when p > 2. No universal pointwise BPB range space can be simultaneously uniformly convex and uniformly smooth unless its dimension is one. We also discuss a version of the pointwise BPB property 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 \N} of norm attaining norm-one operators from X to Y and a point x_0 ∈ X with \|x_0\| = 1, such that \|T_n(x_0)\| \rightarrow 1 but \inf_{n \in \N} \dist (x_0, \{x ∈ X : \|T_n(x)\| = \|x\| = 1 \} > 0. This shows that a version of the Bishop-Phelps-Bollobás property in which the operator is not changed is possible only if one of the involved Banach spaces is one-dimensional.

In this paper, we study conditions assuring that the Bishop-Phelps-Bollobás 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 Y_1 is an absolute summand of Y , then (X, Y_1) has the BPBp;

(b) if X_1 is an absolute summand of X of type 1 or \infty, then (X_1, 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-Bollobás 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.

Some local versions of the Bishop-Phelps-Bollobás property for operators have been recently presented in Dantas et al. (J Math Anal Appl 468(1):304–323, 2018). In the present article, we continue studying these properties for multilinear mappings. We show some differences between the local and uniform versions of these and also provide some interesting examples which show that this study is not just a mere generalization of the linear case. As a consequence of our results, we get that, for 2 < p, q < \infty, the norm of the projective tensor product \ell_p \pten \ell_q is strongly subdifferentiable. 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-Bollobás 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-Bollobás type properties with each other. Some open questions are left throughout the article.

In the first part of our paper, we show that \ell_{\infty} has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as \ell_1(c), also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of c_0(\omega_1) admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler in [7]) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a C^{\infty}-smooth norm. Finally, we prove that there is a proper dense subspace of \ell_{\infty}^c (\omega_1) that admits no Gâteaux smooth norm. (Here, \ell_{\infty}^c(\omega_1) denotes the Banach space of real-valued, bounded, and countably supported functions on \omega_1.)

We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, inherited by one-complemented subspaces, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces. In particular, they are uniformly convex and uniformly smooth, and form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known results on it. Besides, we show that if there is a non-Hilbertian non-separable Banach space with uniform micro-semitransitive (or micro-transitive) norm, then there is a non-Hilbertian separable one. Finally, we show that an L_p(\mu)-space is micro-transitive or uniformly micro-semitransitive only when p = 2.

In this paper, we consider the Bishop-Phelps-Bollobás point property for various classes of operators on complex Hilbert spaces, which is a stronger property than the Bishop-Phelps-Bollobás property. We also deal with analogous problem by replacing the norm of an operator with its numerical radius.

In this paper we are interested to study the set A_{\|\cdot\|} of all norm one linear operators T from X into Y which attain the norm and satisfy the following: given \eps>0, there exists \eta, which depends on \eps and T, such that if \|T(x)\| > 1−\eta, then there is x_0 such that \|x0−x\|< \eps and T itself attains the norm at x_0. We show that every norm one functional on c_0 which attains the norm belongs to A(c_0,\K). Also, we prove that the analogous result is not true neither for A_{\|\cdot\|}(\ell_1,\K) nor A(\ell_{\infty},\K). Under some assumptions, we show that the sphere of the compact operators belongs to A_{\|\cdot\|} and that this is no longer true when some of these hypotheses are dropped. We show also that the natural projections on c_0 or \ell_p, for 1 \leq p < \infty, always belong to this set. The analogous set A_{nu} for numerical radius of an operator instead of its norm is also defined and studied. We give non trivial examples of operators on infinite dimensional Banach spaces which belong to A_{\|\cdot\|} but not to A_{nu} and vice-versa. Finally, we establish some techniques which allow us to connect both sets.

24. The Bishop-Phelps-Bollobás theorem: an overview

**S.D.**, D. García, M. Maestre, and Ó. Roldán

(Universitat Jaume I)

**S.D.**, L.C. García-Lirola, M. Jung, and A. Rueda Zoca

*Quaestiones Mathematicae* (accepted)

(Universitat Jaume I)

**S.D.**, M. Jung, Ó. Roldán, and A. Rueda Zoca

*Mediterranean Journal of Mathematics (accepted)*

(Universitat Jaume I)

**S.D.**, M. Jung, and G. Martínez-Cervantes

*Journal of the Institute of Mathematics of Jussieu (2021)*

(Universitat Jaume I) ✓

**S.D.**, M. Jung, and G. Martínez-Cervantes

*Journal of Mathematical Analysis and Applications (2021)*

(Universitat Jaume I) ✓

**S.D.**, G. Martínez-Cervantes, J.D. Rodríguez-Abellán, and A. Rueda Zoca

*Revista Matemática Iberoamericana (to appear)*

(Universitat Jaume I) ✓

*Banach Journal of Mathematical Analysis (2021)*

(Czech Technical University in Prague)

**S.D.**, G. Martínez-Cervantes, J.D. Rodríguez-Abellán, and A. Rueda Zoca

*RACSAM (2021)*

(Czech Technical University in Prague)

**S.D.**, S.K. Kim, H.J. Lee, and M. Mazzitelli

*The Mathematical Legacy of Victor Lomonosov (2020)*

(Czech Technical University in Prague)

F. Cabello-Sánchez, **S.D.**, V. Kadets, S.K. Kim, H.J. Lee, and M. Martín

*Journal of Mathematical Analysis and Applications (2020)*

(Czech Technical University in Prague)

*Journal of Mathematical Analysis and Applications (2020)*

(Czech Technical University in Prague)

Y.S. Choi, **S.D.**, M. Jung, and M. Martín

*Mediterranean Journal of Mathematics (2019)*

(Czech Technical University in Prague)

**S.D.**, V. Kadets, S.K. Kim, H.J. Lee, and M. Martín

*Linear and Multilinear Algebra (2018)*

(Czech Technical University in Prague)

**S.D.**, V. Kadets, S.K. Kim, H.J. Lee, and M. Martín

*Canadian Journal of Mathematics (2018)*

(Czech Technical University in Prague)

**S.D.**, S.K. Kim, H.J. Lee, and M. Mazzitelli

*Journal of Mathematical Analysis and Applications (2018)*

(Pohang University of Science and Technology)

**S.D.**, D. García, S.K. Kim, U.Y. Kim, H.J. Lee, and M. Maestre

*The Quarterly Journal of Mathematics (2018)*

(Pohang University of Science and Technology)

**S.D.**, D. García, M. Maestre, and M. Martín

*Canadian Journal of Mathematics (2018)*

(Universitat de València)

**S.D.**, D. García, S.K. Kim, H.J. Lee, and M. Maestre

*Linear Algebra and its Applications (2017)*

(Universitat de València)

**S.D.**, S.K. Kim, and H.J. Lee

*Journal of Mathematical Analysis and Applications (2016)*

(Universitat de València)