My hidden notes

Papers under construction

weak-star-KK implies SSD direct proof

Problem 1 in Kadets-Soloviova’s paper on the Bishop-Phelps-Bollobás modulus

Direct proof (BPBp for finite dimensional)

Bollobás complex C(K)

Ideas for FUTURE projects

★ Consider the set of all generating operators which do not attain their norms. Is this set dense in the set of all generating operators?

Mirar las diapositivas de Miguel de la Winter School Korea 2024

★ To find (norm-attaining) operators which belong to the subset BŠ(X,Y) and then study its denseness

The same question can be done for its complement. (BŠ stands for the Bhatia-Šemrl property)

★ To find (norm-attaining) operators which belong to the subset BŠ(X,Y) and then study its denseness

The same question can be done for its complement. (BŠ stands for the Bhatia-Šemrl property)

★ To find (norm-attaining) operators which belong to the subset BŠ(X,Y) and then study its denseness

The same question can be done for its complement. (BŠ stands for the Bhatia-Šemrl property)

★ OH for homogeneous polynomials

In the paper by Geunsu, Mingu and Sun Kwang (THE BIRKHOFF-JAMES ORTHOGONALITY AND NORM ATTAINMENT FOR MULTILINEAR MAPS) the consider multilinear maps already. How about polynomials?

★ NA(J), where J stands for the James space

It is not known that when (X,Y) satisfies the BPBp by assuming that X has the Radon-Nikodým property nor what happens with the particular case of James’ space J (this question was to me by ARZ)

★ Does the BPBp imply the BPBp for finite rank operators?

This question was made by RM to me

Possible directions

10. Counterexample for the density of the symmetric tensors

8. NRA(X) contains no 2 dimensional Banach spaces

7. Elements of tensor product space attaining their Chevet-Saphar norms

6. Possible research lines to work

3. Daugavet points in the homogenous polynomials spaces

1. Phi-properties

↓↓↓ IDEAS FOR THE FUTURE ↓↓↓

w- and w^*-denseness of norm-attaining

Randon-Nikodým property and functions that attain their weighted norms

Open Problems

Strong Subdifferentiability

  • Is it true the following equivalence? For every Banach space X and Y, we have that X \pten Y is reflexive if and only X \pten Y is SSD

Daugavet and delta points

  • Is there a reflexive Banach space X such that P(^N X; \K) is reflexive and contains a Daugavet point? 

Lineability

  • Is it possible to use lineability arguments to construct a very big Y of a nonseparable metric space M such that Y is inside \SNA(M)? 

Lipschitz-free spaces

  • Let T be the Tsirelson’ space. The space F(T) contains \ell_2? How about c_0?
  • The same question for James’ spaces.  

Renorming

  • Does c_0 admite a renorming \|.\| such that (c_0, \|.\|) is a square space?

Daugavet property

  • The weighted Banach space H_v(X,Y) satisfies the Daugavet property?

Temas para estudiantes

Norm-attaining theory

  • Norm-attaining tensors and integral operators 
  • Anti-Bollobás
  • Local properties for compact operators and its relation with SSD