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We show that there are compact linear operators on Banach spaces which cannot be approximated by numerical radius attaining operators. [ABSTRACT FROM AUTHOR]

We study the relation between the intrinsic and the spatial numerical ranges with the recently introduced “approximated” spatial numerical range. As the main result, we show that the intrinsic numerical range always coincides with the convex hull of the approximated spatial numerical range. Besides, we show sufficient conditions and necessary conditions to assure that the approximated spatial numerical range coincides with the closure of the spatial numerical range. [ABSTRACT FROM AUTHOR]

Abstract: We prove that an onto isometry between unit spheres of finite-dimensional polyhedral Banach spaces extends to a linear isometry of the corresponding spaces. [Copyright &y& Elsevier]

Abstract: We study the numerical index of absolute sums of Banach spaces, giving general conditions which imply that the numerical index of the sum is less or equal than the infimum of the numerical indices of the summands and we provide some examples where the equality holds covering the already known case of -, - and -sums and giving as a new result the case of E-sums where E has the RNP and (in particular for finite-dimensional E with ). We also show that the numerical index of a Banach space Z which contains a dense union of increasing one-complemented subspaces is greater or equal than the limit superior of the numerical indices of those subspaces. Using these results, we give a detailed short proof of the already known fact that, for a fixed p, the numerical indices of all infinite-dimensional -spaces coincide. [Copyright &y& Elsevier]

Abstract: We construct an example of a Banach space which is not lush, but whose dual space is lush. This example shows that lushness is not equivalent to numerical index one. A characterization of lushness for some quotient spaces of spaces and new results on C-rich subspaces of (scalar- or vector-valued) spaces are also presented. [Copyright &y& Elsevier]

Abstract: We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality holds for every bounded linear operator . This answers in the positive Question 4.11 of [V. Kadets, M. Martín, J. Merí, Norm equalities for operators on Banach spaces, Indiana Univ. Math. J. 56 (2007) 2385–2411]. More concretely, we show that this is the case of some spaces with few operators constructed in [P. Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004) 151–183] and [G. Plebanek, A construction of a Banach space with few operators, Topology Appl. 143 (2004) 217–239]. We also construct compact spaces and such that and are extremely non-complex, contains a complemented copy of and contains a (1-complemented) isometric copy of . [Copyright &y& Elsevier]

Abstract: We give a lower bound for the polynomial numerical index of order k for real lush spaces. We use this bound to compute the polynomial numerical index of order 2 of the real spaces , and . Finally, we present an example of a real Banach space X whose polynomial numerical indices are positive while the ones of its bidual are zero. [Copyright &y& Elsevier]

Abstract: For a bounded function f from the unit sphere of a closed subspace X of a Banach space Y, we study when the closed convex hull of its spatial numerical range is equal to its intrinsic numerical range . We show that for every infinite-dimensional Banach space X there is a superspace Y and a bounded linear operator such that . We also show that, up to renormig, for every non-reflexive Banach space Y, one can find a closed subspace X and a bounded linear operator such that . Finally, we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range, which we call the Bishop–Phelps–Bollobás property, and which is weaker than the uniform smoothness and the finite-dimensionality. We characterize strong subdifferentiability and uniform smoothness in terms of this property. [Copyright &y& Elsevier]

We introduce a strictly weaker version of the Daugavet property as follows: a Banach space X has this alternative Daugavet property (ADP in short) if the norm identity holds for all rank-one operators T :X→X. In such a case, all weakly compact operators on X also satisfy (). We give some geometric characterizations of the alternative Daugavet property in terms of the space and its successive duals. We prove that the ADP is stable for c0-, l1- and l∞-sums and characterize when some vector-valued function spaces have the property. Finally, we show that a C*-algebra (or the predual of a von Neumann algebra) has the ADP if and only if its atomic projection (respectively, the atomic projection of the algebra) are central. We also establish some geometric properties of JB*-triples, and characterize JB*-triples possessing the ADP and the Daugavet property. [Copyright &y& Elsevier]

We introduce a generalized approximate hyperplane series property for a pair ( X , Y ) of Banach spaces to characterize when ( ℓ 1 ( X ) , Y ) has the Bishop–Phelps–Bollobás property. In particular, we show that ( X , Y ) has this property if X , Y are finite-dimensional, if X is a C ( K ) space and Y is a Hilbert space, or if X is Asplund and Y = C 0 ( L ) , where K is a compact Hausdorff space and L is a locally compact Hausdorff space. [ABSTRACT FROM AUTHOR]

We introduce two Bishop–Phelps–Bollobás moduli of a Banach space which measure, for a given Banach space, what is the best possible Bishop–Phelps–Bollobás theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of these moduli and an inequality with respect to duality. We calculate the two moduli for Hilbert spaces and also present many examples for which the moduli have the maximum possible value (among them, there are spaces and spaces). Finally, we show that if a Banach space has the maximum possible value of any of the moduli, then it contains almost isometric copies of the real space and present an example showing that this condition is not sufficient. [ABSTRACT FROM AUTHOR]

We classify several notions of norm attaining Lipschitz maps which were introduced previously, and present the relations among them in order to verify proper inclusions. We also analyze some results for the sets of Lipschitz maps satisfying each of these properties to be dense or not in Lip 0 (X , Y). For instance, we characterize a Banach space Y with the Radon-Nikodým property in terms of the denseness of norm attaining Lipschitz maps with values in Y. Further, we introduce a property called the local directional Bishop-Phelps-Bollobás property for Lipschitz compact maps, which extends the one studied previously for scalar-valued functions, and provide some new positive results. [ABSTRACT FROM AUTHOR]

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 (μ) space is micro-transitive or uniformly micro-semitransitive only when p = 2. [ABSTRACT FROM AUTHOR]