My hidden notes

Submitted papers (journals)

34. Algebraic genericity of certain families of nets in Functional Analysis

(Submitted on 2024-01-15 to Israel J. Math.)

S.D. and Rodríguez-Vidanes

(Submitted on 2023-11-29 to Adv Math)

S.D., J. Falcó, M. Jung, D. Rodríguez-Vidanes

(Submitted to Proc. A Royal Soc. Edinburgh)

(Submitted on 2024-02-05)

S.D. and R. Medina

 

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

★ Non-norm-attaining tensors + lineability

ARZ already has some results (only one!) in this direction and maybe we could push it further

★ To study integral operators and when do they attain their norms

I don’t know if this would be interesting because integral operators are OPERATORS if you know what I mean…

★ Set points of discontinuity of hyperplanarly continuous maps on Banach spaces

I don’t know if this would be interesting because integral operators are OPERATORS if you know what I mean…

★ To study integral operators and when do they attain their norms

Dani sugiere que miremos lo siguientes artículos:

A continuous tale on continuous and separatey continuous functions

On sets of discontinuous of functions continuous on all lines by Zajicek

Under construction and possible directions

10. Counterexample for the density of the symmetric tensors

S.D., Mingu

9. NA adjoint operators

S.D., Miguel, Mingu (Notes)

“Perhaps the adjoint of every operator from \ell_1 into \ell_1 is norm-attaining”

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

S.D., Dani, Juan, Ella and Martina

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

S.D., Mingu

5. Properties A, B, \alpha, \beta, quasi-\alpha, quasi-\beta for tensor products

S.D., Mingu, Jaagup, LC and Abe

4. Funciones ES en espacios de sucesiones y funciones

S.D., Dani, Juanito

3. Daugavet points in the homogenous polynomials spaces

R. J. Aliaga, S.D., M. Jung, A. Quilis, T. Veoorg

2. On the Bollobás theorem for invariant groups

S.D., J. Falcó, M. Jung, Ó. Roldán

1. Phi-properties

S.D., M. Jung, J.T. Rodríguez, M. Mazzitelli (2022)

↓↓↓ IDEAS FOR THE FUTURE ↓↓↓

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

G. Choi, S.D., M. Jung

Randon-Nikodým property and functions that attain their weighted norms

S.D., R. Medina

Open Problems

Strong Subdifferentiability

  • Is it true that the Banach space P(^2 \c_0, \K) endowed with the sup norm satisfies the sequential w^*-Kadec-Klee property? 
  • 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