The Bishop–Phelps–Bollobás properties in complex Hilbert spaces

Choi, Yun Sung; Dantas, Sheldon; Jung, Mingu

doi:10.1002/mana.201900288

Cited by 3 publications

(5 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Theorem 2, we prove that a space with a micro-transitive norm and the second numerical index strictly positive satisfies the point property for the numerical radius. In particular, real Hilbert spaces satisfy this property, a result that generalizes [17,Theorem 2.5]. We also show that, for Banach spaces with numerical index 1, the point property for the numerical radius is too strong, in the sense that just onedimensional spaces enjoy it (see Proposition 5).…”

supporting

confidence: 55%

“…By uniform, we mean that the η that appears in their definitions depends just on a given ε > 0 (in contrast with the local properties defined in Section 3, η depends on ε and a state, or ε and an operator). It is worth noting that the Bishop-Phelps-Bollobás point property for numerical radius was already introduced by Choi et al in [17] in the context of complex Hilbert spaces. Definition 1.…”

Section: The Point Property For Numerical Radiusmentioning

confidence: 99%

“…Very recently, a stronger property than the BPBp-nu was considered in [17,Theorem 2.5]. Stronger in the sense that if we have T ∈ L(X) with v(T ) = 1 and (x, x * ) ∈ Π(X) satisfying |x * (T (x))| ≈ 1, then the new operator S ∈ L(X) with v(S) = 1 will satisfy |x * (S(x))| = 1 and S ≈ T .…”

mentioning

confidence: 99%

See 2 more Smart Citations

On various types of density of numerical radius attaining operators

Dantas

Kim

Lee

et al. 2023

Linear and Multilinear Algebra

Self Cite

View full text Add to dashboard Cite

In this paper, we are interested in studying Bishop-Phelps-Bollobás type properties related to the denseness of the operators which attain their numerical radius. We prove that every Banach space with a micro-transitive norm and the second numerical index strictly positive satisfies the Bishop-Phelps-Bollobás point property, and we see that the one-dimensional space is the only one with both the numerical index 1 and the Bishop-Phelps-Bollobás point property. We also consider two weaker properties L p,p -nu and L o,o -nu, the local versions of Bishop-Phelps-Bollobás point and operator properties respectively, where the η which appears in their definition does not depend just on ε > 0 but also on a state (x, x * ) or on a numerical radius one operator T . We address the relation between the L p,p -nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and c 0 are examples of Banach spaces satisfying the L p,p -nu, and we exhibit an example of a Banach space with a strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the L o,o -nu and that, if X has a strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality.

show abstract

supporting

confidence: 55%

Section: The Point Property For Numerical Radiusmentioning

confidence: 99%

See 1 more Smart Citation

On various types of density of numerical radius attaining operators

Dantas

Kim

Lee

et al. 2023

Linear and Multilinear Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…In Theorem 2, we prove that a space with micro-transitive norm and second numerical index strictly positive satisfy the point property for the numerical radius. In particular, real Hilbert spaces satisfy this property, a result that generalize [17,Theorem 2.5]. We also show that, for Banach spaces with numerical index 1, both point and operator properties for the numerical radius are too strong, in the sense that just one-dimensional spaces enjoy it (see Proposition 4).…”

supporting

confidence: 55%

“…By uniform, we mean that the η that appears in their definitions depends just on a given ε > 0 (in contrast with the local properties defined in Section 3, where the η depends on ε and a state, or ε and an operator). It is worth noting that the Bishop-Phelps-Bollobás point property for numerical radius was already introduced by Choi et al in [17] in the context of complex Hilbert spaces. Definition 1.…”

Section: The Point and Operator Properties For Numerical Radiusmentioning

confidence: 99%

On various types of density of numerical radius attaining operators

Dantas¹,

Kim²,

Lee³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we are interested in studying two properties related to the denseness of the operators which attain their numerical radius: the Bishop-Phelps-Bollobás point and operator properties for numerical radius (BPBpp-nu and BPBop-nu, respectively). We prove that every Banach space with micro-transitive norm and second numerical index strictly positive satisfy the BPBpp-nu and that, if the numerical index of X is 1, only one-dimensional spaces enjoy it. On the other hand we show that the BPBop-nu is a very restrictive property: under some general assumptions, it holds only for one-dimensional spaces. We also consider two weaker properties, the local versions of BPBpp-nu and BPBop-nu, where the η which appears in their definition does not depend just on ε > 0 but also on a state (x, x * ) or on a numerical radius one operator T . We address the relation between the local BPBpp-nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and c 0 are examples of Banach spaces satisfying the local BPBpp-nu, and we exhibit an example of a Banach space with strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the local BPBop-nu and that, if X has strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality.

show abstract