In this paper we prove some conditions, mostly both necessary and sufficient, for the compactness of the multiplication operator Mm(x)(t): = m(t)x(t) and the substitution operator Sj(x)(t): = x(j(t)) in the function spaces C[0, 1] and BV[0, 1]. More general estimates for the essential norm and the measure of noncompactness of these operators are also obtained. A main emphasis is put on examples and counterexamples which illustrate the abstract results.
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