A note on weak* convergence and compactness and their connection to the existence of the inverse-adjoint. Issue 8 (11th June 2019)

• Record Type: Journal Article
• Authors: Gatica, Gabriel N.
• Is Part Of: Applicable analysis. Volume 98:Issue 8(2019)
• Page Start: 1478
• Page End: 1482
• Publication Date: 2019-06-11
• Subjects: Weak convergence -- weak* convergence -- compactness -- inverse-adjoint; 46B10 -- 46B25 -- 46B28 -- 46B50; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2018.1430780
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: In this note we provide a systematic reasoning to arrive at the reflexivity of the underlying Banach space as a sufficient condition for guaranteeing that any compact operator transforms weak ∗ convergence in strong convergence. Our starting point is an adaptation of the proof for the analogue result holding in the case of the weak convergence. Then, along the way, and as a by-product of the analysis, we characterize the existence of what we call the inverse-adjoint operator.