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In this paper we consider a norm based on the infinitesimal generator of theshift semigroup in a direction. The relevance of such a focus is guaranteed byan abstract representation of a fractional integro-differential operator bymeans of a composition of the corresponding infinitesimal generator. The mainresult of the paper is a theorem establishing equivalence of norms infunctional spaces. Even without mentioning the relevance of this result for theconstructed theory, we claim it deserves to be considered itself.

One more theorem on norm equivalence in the Lebesgue space