doi:10.1007/s43037-020-00114-6engAbadias, L.Bello, G.Yakubovich, D.Functional models up to similarity and a-contractionsART-2021-122483We study the generalization of m-isometries and m-contractions (for positive integers m) to what we call a-isometries and a-contractions for positive real numbers a. We show that an operator satisfying a certain inequality in hereditary form is similar to a-contraction. This improvement of [9, Theorem I] is based on some Banach algebras techniques. We show that our operator classes are closely connected with fractional finite differences. Using this techniques, we get that, given 0 < b< a, an a-contraction need not to be a b-contraction in general, but is a b-contraction if a natural additional requirement is imposed.2021http://zaguan.unizar.es/record/13079810.1007/s43037-020-00114-6http://zaguan.unizar.es/record/130798oai:zaguan.unizar.es:130798info:eu-repo/grantAgreement/ES/DGA-FEDER/E26-17Rinfo:eu-repo/grantAgreement/ES/MINECO-MCIU/PID2019-105979GB-I00Banach Journal of Mathematical Analysis 15, 2 (2021), 34 [29 pp]All rights reservedhttp://www.europeana.eu/rights/rr-f/info:eu-repo/semantics/closedAccess