# Some results on almost L-weakly and almost M-weakly compact operators

## TL;DR

This paper characterizes when semi-compact operators are almost L-weakly or M-weakly compact, linking these properties to the order continuity of the involved Banach space norms and exploring their relation to Dunford-Pettis operators.

## Contribution

It provides necessary and sufficient conditions for semi-compact operators to be almost L-weakly or M-weakly compact, based on the order continuity of Banach space norms.

## Key findings

- Semi-compact operators are almost L-weakly compact iff the target space's norm is order continuous.
- Positive semi-compact operators are almost M-weakly compact iff the dual space's norm is order continuous.
- Relationships between almost L-weakly compact and Dunford-Pettis operators are examined.

## Abstract

In this paper, we present some necessary and sufficient conditions for semi-compact operators being almost L-weakly compact (resp. almost M-weakly compact) and the converse. Mainly, we prove that if $X$ is a nonzero Banach space, then every semi-compact operator $T: X\rightarrow E$ is almost L-weakly compact if and only if the norm of $E$ is order continuous. And every positive semi-compact operator $T:E\rightarrow F$ is almost M-weakly compact if and only if the norm of $E'$ is order continuous. Moreover, we investigate the relationships between almost L-weakly compact operators and Dunford-Pettis (resp. almost Dunford-Pettis) operators.