Strictly singular non-compact operators between Lp spaces

Francisco L. Hernández ^[1] ; Evgeny M. Semenov ^[2] ; Pedro Tradacete ^[3]
1. [1] Universidad Complutense de Madrid
  
  Universidad Complutense de Madrid
  
  Madrid, España
2. [2] Voronezh State University
  
  Voronezh State University
  
  Rusia
3. [3] Instituto de Ciencias Matemáticas
  
  Instituto de Ciencias Matemáticas
  
  Madrid, España
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Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 1, 2023, págs. 181-200
Idioma: inglés
DOI: 10.4171/RMI/1360
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Resumen
- We study the structure of strictly singular non-compact operators between Lp spaces. Answering a question raised in earlier work on interpolation properties of strictly singular operators, it is shown that there exist operators T , for which the set of points .1=p; 1=q/ 2 .0; 1/ .0; 1/ such that T WLp ! Lq is strictly singular but not compact contains a line segment in the triangle ¹.1=p; 1=q/ W 1 < p < q < 1º of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular (i.e., difference of positive) operators defined on subspaces of Lp is also explored.