Cyclically compact operators in Banach spaces

The Boolean-valued interpretation of compactness gives rise to the new notions of cyclically compact sets and operators which deserves an independent study. A part of the corresponding theory is presented in this work. General form of cyclically compact operators in Kaplansky{ Hilbert module as well as a variant ofFredholm Alternative for cyclically compact operators are also given. 1. Preliminaries In this section we present brie y some basic facts about Boolean-valued representations which we need in the sequel. 1.1. Let B be a complete Boolean algebra and let A be a nonempty set. Recall (see [3]) that B(A) denotes the set of all partitions of unity in B with the xed index set A. More precicely, assign n,

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