A New Class of Operators Related to Almost <i>p</i> ‐Limited Sets

ABSTRACT This paper is devoted to the study of almost ‐limited completely continuous (‐lcc) operators, a new class of operators naturally associated with the notion of almost ‐limited sets in Banach lattices. We provide necessary and sufficient conditions under which every positive operator dominated by an ‐lcc operator is also ‐lcc. In addition, we establish necessary conditions under which positive ‐lcc operators admit ‐lcc adjoints, and under which positive operators whose adjoints are ‐lcc are themselves ‐lcc. We also explore the connections between ‐lcc operators and several well‐known classes of operators, including AM‐compact, ‐weakly compact, and ‐weakly compact operators. As an application, we obtain characterizations ensuring that every ‐lcc operator is (weakly) compact.