Search results for "Bishop–Phelps theorem"
showing 5 items of 5 documents
On the Bishop–Phelps–Bollobás theorem for multilinear mappings
2017
Abstract We study the Bishop–Phelps–Bollobas property and the Bishop–Phelps–Bollobas property for numerical radius. Our main aim is to extend some known results about norm or numerical radius attaining operators to multilinear and polynomial cases. We characterize the pair ( l 1 ( X ) , Y ) to have the BPBp for bilinear forms and prove that on L 1 ( μ ) the numerical radius and the norm of a multilinear mapping are the same. We also show that L 1 ( μ ) fails the BPBp-nu for multilinear mappings although L 1 ( μ ) satisfies it in the operator case for every measure μ.
The Bishop–Phelps–Bollobás theorem for operators
2008
AbstractWe prove the Bishop–Phelps–Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop–Phelps–Bollobás theorem holds for operators from ℓ1 into Y. Several examples of classes of such spaces are provided. For instance, the Bishop–Phelps–Bollobás theorem holds when the range space is finite-dimensional, an L1(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.
The Bishop–Phelps–Bollobás point property
2016
Abstract In this article, we study a version of the Bishop–Phelps–Bollobas property. We investigate a pair of Banach spaces ( X , Y ) such that every operator from X into Y is approximated by operators which attain their 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–Bollobas 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 l 1 and l ∞ sums of Banach spaces and we also study this property for bilinear mappings.
Some kind of Bishop-Phelps-Bollobás property
2016
In this paper we introduce two Bishop–Phelps–Bollobas 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 of 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 of property 1 for the pairs (lp,lq) is also provided.
A group invariant Bishop-Phelps theorem
2021
We show that for any Banach space and any compact topological group G ⊂ L ( X ) G\subset L(X) such that the norm of X X is G G -invariant, the set of norm attaining G G -invariant functionals on X X is dense in the set of all G G -invariant functionals on X X , where a mapping f f is called G G -invariant if for every x ∈ X x\in X and every g ∈ G g\in G , f ( g ( x ) ) = f ( x ) f\big (g(x)\big )=f(x) . In contrast, we show also that the analog of Bollobás result does not hold in general. A version of Bollobás and James’ theorems is also presented.