A note on weak* convergence and compactness and their connection to the existence of the inverse-adjoint

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${}^{\ast }$ 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.

KEYWORDS:

• Weak convergence
• weak* convergence
• compactness
• inverse-adjoint

AMS SUBJECT CLASSIFICATIONS:

• 46B10
• 46B25
• 46B28
• 46B50

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, and by Centro de Investigación en Ingeniería Matemática (CI${}^2$MA), Universidad de Concepción.