: //matematika.etf.bg.ac.yu A NOTE ON COMPACT OPERATORS

Let H be a separable complex Hilbert space and let B(H) be the algebra of bounded linear operators on H. Recall that an operator T ∈ B(H) is said to be compact if for every bounded sequence {xn} of vectors in H, the sequence {Txn} contains a converging subsequence. An operator T is said to be of finite rank if the range of T, R(T) is finite dimensional. It is easily seen that every finite rank operator is compact, however, the converse is false. The operator T is said to be of almost finite rank of T is the limit, in the norm topology of B(H), of a sequence of finite rank operators. Finally, the operator T is said to be completely continuous (or C.C.) if for every weakly convergent sequence {xn}, the sequence {Txn} converges. A sequence {xn} in H is said to converge weakly if the sequence {〈xn, y〉} of numbers converges for all y. For these concepts see [1], [2], [3], [6]. The following result is well known (see [4], [7]): Theorem. Let T ∈ B(H). The following statements are equivalent: 1. T is compact. 2. T is of almost finite rank. 3. T is completely continuous. In this note we introduce the concepts of a quasi-compact operator and semi-compact operator and we show the equivalence of these concepts with compactness