74 research outputs found
A note on numerical radius attaining mappings
We prove that if every bounded linear operator (or -homogeneous
polynomials) with the compact approximation property attains its numerical
radius, then is a finite dimensional space. Moreover, we present an
improvement of the polynomial James' theorem for numerical radius proved by
Acosta, Becerra Guerrero and Galn in 2003. Finally, the denseness of
weakly (uniformly) continuous -homogeneous polynomials on a Banach space
whose Aron-Berner extensions attain their numerical radii is obtained.Comment: 15 page
A generalized ACK structure and the denseness of norm attaining operators
Inspired by the recent work of Cascales et al., we introduce a generalized
concept of ACK structure on Banach spaces. Using this property, which we call
by the quasi-ACK structure, we are able to extend known universal properties on
range spaces concerning the density of norm attaining operators. We provide
sufficient conditions for quasi-ACK structure of spaces and results on the
stability of quasi-ACK structure. As a consequence, we present new examples
satisfying the (Lindenstrauss) property B, which have not been known
previously. We also prove that property B is stable under injective tensor
products in certain cases. Moreover, ACK structure of some Banach spaces of
vector-valued holomorphic functions is also discussed, leading to new examples
of universal BPB range spaces for certain operator ideals.Comment: 17 page
The Daugavet and Delta-constants of points in Banach spaces
We introduce two new notions called the Daugavet constant and
-constant of a point, which measure quantitatively how far the point is
from being Daugavet point and -point and allow us to study Daugavet and
-points in Banach spaces from a quantitative viewpoint. We show that
these notions can be viewed as a localized version of certain global
estimations of Daugavet and diametral local diameter two properties such as
Daugavet indices of thickness. As an intriguing example, we present the
existence of a Banach space in which all points on the unit sphere have
positive -constants despite the Daugavet indices of thickness of
being zero. Moreover, using the Daugavet and -constants of points in
the unit sphere, we describe the existence of almost Daugavet and
-points as well as the set of denting points of the unit ball. We also
present exact values of the Daugavet and -constant on several classical
Banach spaces, as well as Lipschitz-free spaces. In particular, it is shown
that there is a Lipschitz-free space with a -point which is the
furthest away from being a Daugavet point. Finally, we provide some related
stability results concerning the Daugavet and -constant.Comment: 35 pages, 3 figure
Norm-attaining operators which satisfy a Bollobás type theorem
This is a pre-print of an article published in Banach Journal of Mathematical Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s43037-020-00113-7In this paper, we are interested in studying the set A(parallel to center dot parallel to) (X, Y) of all norm-attaining operators T from X into Y satisfying the following: given epsilon > 0, there exists eta such that if parallel to Tx parallel to > 1 - eta, then there is x(0) such that parallel to x(0) - x parallel to < 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 A(parallel to center dot parallel to) (c(0), K). Also, we prove that the analogous result holds neither for A(parallel to center dot parallel to) (l(1), K) nor A(parallel to center dot parallel to) (l(infinity), K). Under some assumptions, we show that the sphere of the compact operators belongs to A(parallel to center dot parallel to) (X, Y) and that this is no longer true when some of these hypotheses are dropped. The analogous set 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 A(parallel to center dot parallel to) (X, X) and A(nu)(X) when X = c(0) or l(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 A(parallel to center dot parallel to) (X, X) but not to A(nu)(X) and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums
The Bishop-Phelps-Bollobás properties in complex Hilbert space
In this paper, we consider theBishop–Phelps–Bollobás point propertyfor variousclasses of operators on complex Hilbert spaces, which is a stronger property thanthe Bishop–Phelps–Bollobás property. We also deal with analogous problem byreplacing the norm of an operator with its numerical radius
On the existence of non-norm-attaining operators
In this paper we provide necessary and sufficient conditions for the
existence of non-norm-attaining operators in . By using a
theorem due to Pfitzner on James boundaries, we show that if there exists a
relatively compact set of (in the weak operator
topology) such that is an element of its closure (in the weak operator
topology) but it is not in its norm closed convex hull, then we can guarantee
the existence of an operator which does not attain its norm. This allows us to
provide the following generalization of results due to Holub and Mujica. If
is a reflexive space, is an arbitrary Banach space, and the pair
has the bounded compact approximation property, then the following are
equivalent:
(i) ;
(ii) Every operator from into attains its norm;
(iii) ;
where denotes the topology of compact convergence. We conclude the
paper presenting a characterization of the Schur property in terms of
norm-attaining operators
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