Weak-star quasi norm attaining operators

For Banach spaces X and Y, a bounded linear operator T : X -> Y* is said to weak-star quasi attains its norm if the sigma(Y*, Y)-closure of the image by T of the unit ball of X intersects the sphere of radius ||T|| centered at the origin in Y *. This notion is inspired by the quasi-norm attainment of operators introduced and studied in [5]. As a main result, we prove that the set of weak-star quasi norm attaining operators is dense in the space of bounded linear operators regardless of the choice of the Banach spaces, furthermore, that the approximating operator can be chosen with additional properties. This allows us to distinguish the properties of weak-star quasi norm attaining operators from those of quasi norm attaining operators. It is also shown that, under certain conditions, weak-star quasi norm attaining operators share numbers of equivalent properties with other types of norm attaining operators, but that there are also a number of situations in which they behave differently from the others.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper studies weak-star quasi norm attaining operators and proves that the set of such operators is dense in the space of bounded linear operators regardless of the choice of Banach spaces. It is also shown that weak-star quasi norm attaining operators have distinct properties from other types of norm attaining operators, although they may share some equivalent properties under certain conditions.