This chapter focuses on the compactness properties of positive operators. A compact operator sends an arbitrary norm bounded sequence onto a sequence with a norm convergent subsequence. For this reason, when operators are associated with integral equations, the compact operators are the most desirable. Besides being compact, an operator can enjoy a number of other compactness properties. An operator with some type of compactness is more useful than an arbitrary operator. This chapter studies various compactness properties of operators on Banach spaces. The chapter deals with compact operators, weakly compact operators, L- and M-weakly compact operators, and Dunford–Pettis operators. Particular emphasis is given to compactness properties of a positive operator dominated by a compact operator. The relationships between the ring and order ideals generated by a positive operator are examined. © 1985, Academic Press Inc.
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