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We characterize mappings Si and Ti, not necessarily linear, from sets $\mathcal {J}_{i}$, i=1,2, onto multiplicative subsets of function algebras, subject to the following conditions on the peripheral spectra of their products: σπ(S1(a)S2(b))⊂σπ(T1(a)T2(b)) and σπ(S1(a)S2(b))∩σπ(T1(a)T2(b))≠∅, $a\in \mathcal {J}_{1}$, $b\in \mathcal {J}_{2}$. As a direct consequence we obtain a large number of previous results about mappings subject to various spectral conditions.

Mappings onto multiplicative subsets of function algebras and spectral properties of their products