Functional models up to similarity and a-contractions

We study the generalization of m-isometries and m-contractions (for positive integers m) to what we call a-isometries a-contractions for real numbers a. show that an operator satisfying a certain inequality in hereditary form is similar a-contraction. This improvement [9, Theorem I] based on some Banach algebras techniques. our classes are closely connected with fractional finite differences. Using this techniques, get that, given $$0<b<a$$ , a-contraction need not be b-contraction general, but if natural additional requirement imposed.