Nice operators into $$L_1$$ L 1 -preduals

Cabrera-Serrano, Ana M.; Mena-Jurado, J. F.

doi:10.1007/s13163-016-0219-9

Cited by 3 publications

(2 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While the characterization of extreme contractions defined between Hilbert spaces is well known [5,9,18,20], the characterization of the same is still elusive, in the general setting of Banach spaces. There are several papers including [1,2,3,4,8,11,12,13,15,19,22,23,24], that deal with the study of extreme contractions of operators defined between some special Banach spaces. In these papers, the authors have studied extreme contractions defined between particular Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

Characterization of extreme contractions through $k-$smoothness of operators

Mal¹,

Paul²,

Dey³

2020

Preprint

View full text Add to dashboard Cite

We characrterize extreme contractions defined between finitedimensional polyhedral Banach spaces using k-smoothness of operators. We also explore weak L-P property, a recently introduced concept in the study of extreme contractions. We obtain a sufficient condition for a pair of finitedimensional polyhedral Banach spaces to satisfy weak L-P property. As an application of these results, we explicitly compute the number of extreme contractions in some special Banach spaces. Our approach in this paper in studying extreme contractions lead to the improvement and generalization of previously known results.

show abstract

Section: Introductionmentioning

confidence: 99%

Characterization of extreme contractions through $k-$smoothness of operators

Mal¹,

Paul²,

Dey³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The study of isometric properties of the space L(X, Y ) of all bounded linear operators between two Banach spaces X and Y and their impact on the domain and range spaces is a traditional subject of Banach space theory, and it remains to be an active area of research. For instance, in the second part of twentieth century there were a number of results [5,22,40,47,48,49,50] on the structure of extreme points of the unit ball of L(X, Y ) (sometimes known as extreme operators or extreme contractions), but the subject attracts researchers till now, see for instance [10,11,32,41,46] and references therein. When X = Y , the space L(X) := L(X, X) is a Banach algebra with unit Id (or Id X if it is necessary to mention), and there are many deep results in this case (see, for instance, the classical references [42,45]).…”

Section: Introductionmentioning

confidence: 99%

On the numerical index with respect to an operator

Kadets¹,

Martín²,

Merí³

et al. 2019

Preprint

View full text Add to dashboard Cite

The aim of this paper is to study the numerical index with respect to an operator between Banach spaces. Given Banach spaces X and Y , and a norm-one operator G ∈ L(X, Y ) (the space of all bounded linear operator from X to Y ), the numerical index with respect to G, n G (X, Y ), is the greatest constant k 0 such thatHere, we first provide some tools to study the numerical index with respect to G.Next, we present some results on the set N (L(X, Y )) of the values of the numerical indices with respect to all norm-one operators on L(X, Y ). For instance, we show that N (L(X, Y )) = {0} when X or Y is a real Hilbert space of dimension greater than one and also when X or Y is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. In the real case, we show that forfor all real Banach spaces X and Y , where Mp = sup t∈[0,1] |t p−1 −t| 1+t p . For complex Hilbert spaces H 1 , H 2 of dimension greater than one, we show that N (L(H 1 , H 2 )) ⊆ {0, 1/2} and the value 1/2 is taken if and only if H 1 and H 2 are isometrically isomorphic. Besides, N (L(X, H)) ⊆ [0, 1/2] and N (L(H, Y )) ⊆ [0, 1/2] when H is a complex infinite-dimensional Hilbert space and X and Y are arbitrary complex Banach spaces. We also show that N (L(L 1 (µ 1 ), L 1 (µ 2 ))) ⊆ {0, 1} and N (L(L∞(µ 1 ), L∞(µ 2 ))) ⊆ {0, 1} for arbitrary σ-finite measures µ 1 and µ 2 , in both the real and the complex cases. Also, we show that the Lipschitz numerical range of Lipschitz maps from a Banach space to itself can be viewed as the numerical range of convenient bounded linear operators with respect to a bounded linear operator. Further, we provide some results which show the behaviour of the value of the numerical index when we apply some Banach space operations, as constructing diagonal operators between c 0 -, 1 -, or ∞-sums of Banach spaces, composition operators on some vector-valued function spaces, taking the adjoint to an operator, and composition of operators.

show abstract