The ideal of $p$-compact operators: a tensor product approach

Daniel Galicer ¹   ; Silvia Lassalle ¹   ; Pablo Turco ¹

¹ Departamento de Matemática – Pab. I Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires (1428) Buenos Aires, Argentina

Studia Mathematica, Tome 211 (2012) no. 3, pp. 269-286

Résumé

We study the space of $p$-compact operators, $\mathcal K_p$, using the theory of tensor norms and operator ideals. We prove that $\mathcal K_p$ is associated to $/d_p$, the left injective associate of the Chevet–Saphar tensor norm $d_p$ (which is equal to $g_{p'}'$). This allows us to relate the theory of $p$-summing operators to that of $p$-compact operators. Using the results known for the former class and appropriate hypotheses on $E$ and $F$ we prove that $\mathcal K_p(E;F)$ is equal to $\mathcal K_q(E;F)$ for a wide range of values of $p$ and $q$, and show that our results are sharp. We also exhibit several structural properties of $\mathcal K_p$. For instance, we show that $\mathcal K_p$ is regular, surjective, and totally accessible, and we characterize its maximal hull $\mathcal K_p^{\max}$ as the dual ideal of $p$-summing operators, $\varPi_p^{\rm dual}$. Furthermore, we prove that $\mathcal K_p$ coincides isometrically with $\mathcal {QN}_p^{\rm dual}$, the dual to the ideal of the quasi $p$-nuclear operators.

Détail

DOI : 10.4064/sm211-3-8

Keywords: study space p compact operators mathcal using theory tensor norms operator ideals prove mathcal associated injective associate chevet saphar tensor norm which equal allows relate theory p summing operators p compact operators using results known former class appropriate hypotheses prove mathcal f equal mathcal f wide range values results sharp exhibit several structural properties mathcal instance mathcal regular surjective totally accessible characterize its maximal hull mathcal max dual ideal p summing operators varpi dual furthermore prove mathcal coincides isometrically mathcal dual dual ideal quasi p nuclear operators