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We prove a general type description result for the multipliers acting betweentwo periodic Bessel potential spaces, defined on the $n$--dimensional torus, ina case when their smoothness indices are of different signs. This is donethrough the detailed examination of a periodic analogue of the linear operator$J_s$, which is employed in the definition of the scale of the Bessel potentialspace defined on the whole space $\mathbb{R}^n$. Our method of defining thisperiodic analogue of $J_s$ uses the results about an asymptotic behaviour ofthe generalized Fourier coefficients and existence of a natural homeomorphismbetween the spaces $\mathcal{D}'(\mathbb{T}^n)$ and $S'_{2 \cdot\pi}(\mathbb{R}^n)$, where the latter consists of all $2 \cdot \pi$--periodicdistributions from the dual Schwartz space $\mathcal{S}'(\mathbb{R}^n)$.

Multipliers in the scale of periodic Bessel potential spaces with smoothness indices of different signs