  We 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 any Hilbert space operator, satisfying
an inequality of certain class (in hereditary form), is similar to
$a$-contractions. This result is based on some Banach algebras techniques and
is an improvement of a recent result by the last two authors. We also prove
that any $a$-contraction $T$ is a $b$-contraction, if $b<a$ and one imposes an
additional condition on the growth of the norms of $T^n x$, where $x$ is an
arbitrary vector. Here we use some properties of fractional finite differences.
