Positively p-nuclear operators, positively p-integral operators and approximation properties

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 26; no. 1
• Main Authors: Chen, Dongyang, Belacel, Amar, Chávez-Domínguez, Javier Alejandro
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-02-2022; Springer Nature B.V
• Subjects: Approximation; Calculus of Variations and Optimal Control; Optimization; Econometrics; Fourier Analysis; Mathematical analysis; Mathematics; Mathematics and Statistics; Operator Theory; Operators; Potential Theory; Approximation properties; 46B28; nuclear operators; Positively; Latticially; 46B45; Primary 47B10; 46B42; integral operators
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-022-00865-6

In the present paper, we introduce and investigate a new class of positively
p
-nuclear operators that are positive analogues of right
p
-nuclear operators. One of our main results establishes an identification of the dual space of positively
p
-nuclear operators with the class of positive
p
-majorizing operators that is a dual notion of positive
p
-summing operators. As applications, we prove the duality relationships between latticially
p
-nuclear operators introduced by O. I. Zhukova and positively
p
-nuclear operators. We also introduce a new concept of positively
p
-integral operators via positively
p
-nuclear operators and prove that the inclusion map from
L
p
∗
(
μ
)
to
L
1
(
μ
)
(
μ
finite) is positively
p
-integral. New characterizations of latticially
p
-integral operators and positively
p
-integral operators are presented and used to prove that an operator is latticially
p
-integral (resp. positively
p
-integral) precisely when its second adjoint is. Finally, we describe the space of positively
p
-integral operators as the dual of the
‖
·
‖
Υ
p
-closure of the subspace of finite rank operators in the space of positive
p
-majorizing operators. Approximation properties, even positive approximation properties, are needed in establishing main identifications.